Topological Spaces and Groups: part 3: Fast Review

Reference: Topological Transformation Groups by Deane Montgomery and Leo Zippin, Dover Publications, available Amazon India.

1.21 COSET SPACE of LOCAL GROUPS

DEFINITION. A closed subset H of a local group G is called a subgroup if a neighbourhood V of e exists such that V^{2} is defined, V open and symmetric, and:

i) x \in H \bigcap V implies x^{-1} \in H \bigcap V

ii) x, y \in H \bigcap V implies xy \in H.

The subgroup is called invariant if a V exists satisfying (i) and (ii) and also

(iii) y \in V, h \in H \bigcap V implies y^{-1}ky \in H

LEMMA.

Let G be a local group, H a subgroup, V a neighbourhood as described above, and let W be a symmetric open neighbourhood of a e with W^{16} \subset V.. Then for x, y \in W and x^{-1}y \in H is an equivalence relation and x^{-1}y \in H if and only if xH \bigcap W = yH \bigcap W.

Proof:

The relation x^{-1}y \in H is reflexive since x^{-1}x = e \in H. It is symmetric since if x, y \in W then x^{-1}y \in W^{2} hence y^{-1}x is defined and belongs to W^{2}. Now x^{-1}y \in H implies y^{-1}x \in H. The relation is transitive because x, y, z \in W and x^{-1}y , y^{-1}z \in H implies x^{-1}z \in H by condition (2).

It can be seen that the sets xH \bigcap W where x \in W are the equivalence classes of this equivalence relation. They are called the local cosets and are the points of a coset space W/(H \bigcap W). We shall now define the topology for this space.

Let T denote the natural map of W onto W/(H \bigcap W) defined by

T : x \rightarrow (xH \bigcap W)

Note that T (for local groups) is defined only on W. Let the open sets of the coset space be those which have an open set in W as inverse under T. Then T is open and continuous.

COROLLARY.

With the same assumptioins and notation as in the preceding Lemma, let H be invariant. Then W/(H \bigcap W) is a local group when the product is defined in the natural way.

The product function is defined for x, y \in W by:

(xH \bigcap W) (yH \bigcap W) = (xyH \bigcap W)

To prove that this is a single valued function (or a well-defined function) on cosets it needs to be shown that for h, h^{'} \in H and xh, yh^{'} \in W the product (xy)^{-1}(xhyh^{'}) exists and is in H. The product is in W^{4} and defines an element of V, This element can be seen to be in H \bigcap V by the use of the associative law and condition (3) above; it must be verified that all indicated products are well-defined. Finally let U_{0} be an open neighbourhood of e in G, with U_{0}^{2} \subset W. It can be seen that the product of each pair of elements in the open set T(U_{0}H \bigcap W) is defined as an element of W/(H \bigcap W). The remaining details can be verified by the reader. ( I have yet to ! :-))

THEOREM.

A local group G possesses such a family of neighbourhoods as is defined in Section 1.17 (previous blog). For every subgroup H of G and neighbourhood W of e (this neighbourhood chosen as above) the coset space W/(W \bigcap H) is a Hausdorff space.

The proof is the same as in 1.16 and 1.17. (previous blog)

Remarks on the side: Some of the principal results of the succeeding chapters are valid for local groups as well as global groups. However, the consideration of local groups in each preliminary Lemma and Theorem is not feasible in a work of this kind. In the sequel we shall only occasionally need to make explicit mention of the local groups.

1.22 INVARIANT METRIC

The construction of metrics and invariant metrics in groups was carried out by Garrett Birkhoff and Kakutani independently.

A local group is called metric if some neighbourhood of the identity is metric.

THEOREM.

Let G be a topological group whose open sets at e have a countable basis. Then G is metrizable (Section 1.9) and moreover, there exists a metric which is right invariant.

(From Section 1.9 we reproduce the definition of metrizable: A space is called metrizable if a metric can be defined for it which induces in it the original topology.)

Let U_{i} for i=1,2, \ldots be a countable basis for open sets at e and for each positive integer n let O_{n}=\bigcap_{1}^{n}U_{i}. The sets O_{n} are monotonic decreasing and form a basis at e. Let V_{r} for each dyadic rational r=k/2^{n}, where 1 \leq k \leq 2^{n}, where n \in \mathcal{N} be a family of neighbourhoods of e in G such as is constructed in Section 1.17 (previous blog).

Define a function

f(x,y) on G as follows:

(i) f(x,y) =0 if and only if y \in V_{r}V_{r}^{-1}x for every r.

(ii) otherwise f(x,y) =lub(r) where y \in V_{r}V_{r}^{-1}x

For any set U and element a in G, y \in Ux if and only if ya \in Uxa. From this it follows that f(x,y) is right invariant:

(iii) f(x,y) = f(xa,ya)

Next from the fact that V_{r}V_{r}^{-1} is symmetric it follows that xy^{-1} \in V_{r}V_{r}^{-1} if and only if yx^{-1} \in V_{r}V_{r}^{-1} for the same r. It follows that f is symmetric

f(x,y) = f(y,x)

The sets V_{1/2^{n}} are symmetric so that V_{1/2^{n}}V_{1/2^{n}}^{-1} = V_{1/2^{n}}^{2} \subset V_{1/2^{n-1}} by (5) of section 1.17. Also V_{1/2^{n-1}} \subset O_{n-1} and \bigcap O_{n-1}=e. It follows now that

f(x,y)=0 if and only if x=y

We next define the distance function:

*) d(x,y) = lub_{u}|f(x,u)-f(y,u)|

The definition shows that d(x,y) = d(y,x)

d(x,y) \geq f(x,y) \geq 0 and d(x,x)=0

If d(x,y)=0 then f(x,y)=0 and x=y. Finally the triangle inequality:

d(x,z) =lub_{u}|f(x,u) -f (y,u) + f(y,u)-f(z,u) | \leq d(x,y) + d(y,z)

The right invariance of the metric is shown as follows:

d(xa, ya ) = lub_{u}|f(xa,u) - f(ya,u)| = lub_{va}|f(xa,va) - f(ya,va)| = lub_{v}|f(x,v)-f(y,v)|=d(x,y)

Finally we must show the equivalence of the original neighbourhoods of G and the sphere neighbourhoods of the metric. It is sufficient to verify this at a e because of the invariance of the metric on the one hand and the translation properties of G (Section 1.13 previous blog) on the other. Let S_{1/2^{n}} denote the set of elements whose distance from a is less than 1/2^{n}. The fact that

V_{1/2^{n+1}} \subset S_{1/2^{n}}

shows that the metric spheres are neighbourhoods of e in the original topology. It remains to show that they form a basis at e. Let a neighbourhood O of e be given. Since the sets O_{n} are a basis at e there is an integer k such that O_{k} \subset O. Then if x \in S_{1/2^{k+1}}, and d(e,x) < 1/2^{k+1} and f(e,x) < 1/2^{k+1}, and finally

x \in V_{1/2^{k+1}}^{2} \subset V_{1/2^{k}}\subset O_{k} hence

S_{1/2^{k+1}}\subset O_{k}

This completes the proof.

1.23 METRIC HOMOGENEOUS SPACE

THEOREM.

Let H be a closed subgroup of a metric group G. Then G/H is metrizable.

Proof.

Let d(x,y) be a right invariant metric in G. Define D(xH, yH) as follows:

D(xH, yH) = glb \hspace{0.1in} d(xa, yb) where a, b \in H

then for every \epsilon >0, H contains elements a, b, c, d such that

D(xH, yH) + D(yH, zH) +2\epsilon \geq d(xa,yb) + d(yc, zd) = d(xa,yb) + d(yb,zdc^{-1}b) \geq d(xa,zdc^{-1}b) \geq D(xH,zH)

This shows that D satisfies the triangle inequality. It can be verified that D is single-valued in G/H x G/H, that is , D is symmetric and that D(xH,yH) =0 if and only if xH=yH. Then D is a metric.

The set yH belongs to the set S_{1/2^{n}}(e).xH if D(xH, yH) < 1/2^{n}. Therefore any neighbourhood of xH in the original topology includes a metric neighbourhood. The converse can also be shown. Therefore the metric D induces the topology of G/H.

1.24 CONNECTEDNESS

A subset M of a topological space S is called connected if whenever M = A \bigcup B where A and B are open relative to M and are not empty, it follows that A \bigcap B is not empty.

LEMMA.

If M is connected and A is a subset which is both open and closed relative to M, then A = M, or A is the empty set.

This follows from the definition.

LEMMA.

If G is a topological group and H is a subgroup which is open, then H is also closed.

Since H is open each coset of H is open. The compliment of H is the union of cosets of H. Hence, the complement of H is open so H must be closed.

THEOREM.

If G is a connected group and W is an open neighbourhood of e, then G = \bigcup_{n}W^{n}.

The set H  = \bigcup_{n}(W \bigcap W^{-1})^{n} is an open subgroup of G. Therefore H is a closed subgroup, and then it follows from the connectedness of G that H=G.

A union of an arbitrary number of connected subsets is connected, provided every two of the sets have a point in common. It follows that to each point x \in S there is uniquely associated a maximal connected subset of M containing x; these maximal connected subsets are called components of S. The closure of a component is connected — therefore the component is closed. A space is called totally-disconnected when each point if a component.

LEMMA.

If M is connected and f is a continuous map of M into a space N then f(M) is connected.

If f(M) = A \bigcup B and A \bigcap B=\Phi

Then M = f^{-1}(A) \bigcup f^{-1}(B) and f^{-1}(A) \bigcap f^{-1}(B) = \Phi

COROLLARY.

If G is a connected group and H is a closed subgroup, then G/H is a connected space.

DEFINITION.

A space is called normal if any two disjoint closed sets are contained in disjoined open sets; two sets are disjoined (disjunct, mutually separated) if they have no common points.

LEMMA.

A compact Hausdorff space is normal.

The proof is straight forward and is omitted.

LEMMA.

Let S be a compact Hausdorff space and x a point in S. Let Q be a set of indices \{ q\}. If \{ A_{q}\} is the set of all compact open subsets containing x, then C = \bigcap_{a}A_{a} is the x-component of S.

Let

\bigcap_{a}A_{a} = X \bigcup Y where X \bigcap Y = \Phi

and X and Y are closed. There exist open sets V \supset X and W \supset Y such that V \bigcap W = \Phi. Hence,

*) (S - (V \bigcup W)) \bigcap (\bigcap _{q}A_{q}) = \Phi

and there is a finite set of indices Q^{'} \subset Q such that *) continues to hold if the intersection \bigcap_{a} os restricted to q \in Q^{'}. Let A = \bigcap_{q^{'}}A_{q^{'}} where q^{'} \in Q^{'}. Then, A is compact and open and x \in A.

Finally, since

A = (A \bigcap V) \bigcup (A \bigcap W)

it follows that A \bigcap V and A \bigcap W are compact open sets only one of which can contain x. Since the system \{ A_{q}\} where q \in Q is maximal, one of A \bigcap V and A \bigcap W is in \{ A_{q}\}, say A \bigcap V \in \{ A_{q}\} and therefore \bigcap_{a}A_{a} \subset A \bigcap V \subset V; hence, Y is empty.

This shows that C = \bigcap_{q}A_{q} is connected. It is now clear that Cis the component containing x.

COROLLARY.

Let S be compact Hausdorff space, C be a component of S, let F be a closed set and suppose that F \bigcap C = \Phi. Then there is a compact open set A^{'}, C \subset A^{'}, F \bigcap A^{'} = \Phi.

It follows from the hypothesis that there is a finite set Q^{''} \subset Q such that

F \bigcap (\bigcap_{q^{''}}A_{q^{''}}) = \Phi where q^{''} \in Q^{''}

Let

A^{'} = \bigcap_{a^{''}}A_{q^{''}}.

This is the set required by the corollary.

1.25 THE IDENTITY COMPONENT G_{0}

THEOREM.

Let g be a topological group and G_{0} the component of G containing the identity. Then G_{0} is closed invariant subgroup and the factor group G/G_{0} is totally disconnected.

If M is a connected subset of G, M^{-1} is connected, Mx and xM where x \in G are connected (section 1.13). It follows that the identity component G_{0} is a group and that the cosets xG_{0} are also components. We have already remarked that components are closed sests. Then G_{0} is a closed subgroup. Since x^{-1}G_{0}x is connected and contains the identity e for every x \in G, it is clear that G_{0} is invariant.

Now let M denote a connected subset of G/G_{0}. Let T be the natural map of G onto G/G_{0}. We shall show that T^{-1}(M) is connected, thus proving that it is in a coset of G/G_{0}. Suppose that

T^{-1}(M) = A \bigcup B

where A and B are relatively open (and closed ) in T^{-1}(M) and A \bigcap B is empty. Then T(T^{-1}(M)) = M = T(A) \bigcup T(B). Since there exists an open set, say U, in G, such that A = U \bigcap T^{-1}(M) and since T is an open set, say U, in G, such that A = U \bigcap T^{-1}(M) and since T is an open map, the set T(A) is open relatively to M. This is true also of T(B). Since M is connected it follows that T(A) \bigcap T(B) is not empty. Let xG_{0} \in T(A) \bigcap T(B). Since xG_{0} is a connected subset of G,

xG_{0} = (xG_{0} \bigcap A) \bigcup (xG_{0} \bigcap B

leads to a contradiction.

A subgroup of a group G is called central if each of its elements communtes with every element in the whole group. Subgroups H and g^{-1}Hg are called conjugate.

THEOREM.

If G is a connected group and H is an invariant totally disconnected subgroup then H is central.

Let x be an element of H and consider the map of G into H depending on x:

g \rightarrow gxg^{-1} where g \in G.

Since H contains no connected set with more than one point, the image of G which contains x, must coincide with x. This completes the proof.

1.26 TRANSFORMATION GROUPS

We shall soon confine our attention to groups which are locally compact, and we shall be particularly interested in the transformation groups of locally-euclidean spaces. For the time being, we continue to study the more general phenomenon.

DEFINITION I.

Let M denote a Hausdorff space and G a topological group each element of which is a homeomorphism of M onto itself:

(1) f(g; x) = g(x) = x^{'} \in M; g \in G, and x \in M

The pair (G,M), or sometimes G itself will be called a topological transformation group if for every pair of elements of G, and every x of M,

(2) g_{1}(g_{2}(x)) = (g_{1}g_{2})(x) ;

and if x^{'} = g(x) = f(g; x) is continuous simultaneously in x \in M and g \in G.

From (2) and from the fact that each g is one-one on M, it follows that for every x \in M

(3) e(x)=x; e the identity in G.

If e is the only element in G which leaves all of M fixed, that is if e is the only element satisfying (3) for all x, then G is called effective.

DEFINITION I’

A pair (G,M) will be called a local transformation group if all conditions of the preceding Definition are fulfilled excepting only that G is assumed to be a local group, and condition (2) holds whenever g_{1}g_{2} is defined.

Some of the remarks below apply both to local and global case for the most part, except when noted.

Let G be a transformation group, g \in G, x, y \in M and suppose that

g(x) = y \neq x

Since M is a Hausdorff space, there is a neighbourhood Y of y not containing x. By the definition of the transformation group, there is a neighbourhood U of g such that for g^{'} \in U, g^{'}(x) \in Y. Therefore, g^{'}(x) \neq x, and it follows that the set of elements of G which leave x fixed is closed. Therefore this set is a closed subgroup of G. We shall denote it by G_{x}. Similarly, the set of elements of G leaving fixed every point of M is a closed subgroup of G_{M}.

THEOREM

Let (G,M) be a transformation group, and let K be the closed subgroup of G leaving all of M fixed. Then K is invariant and G/K is an effective transformation group of M under the action:

T^{*}: (gK)(x) = g(x) where g \in G.

For x \in M, h \in K, g \in G we must always have: (g^{-1}hg)(x)=x. This shows the invariance of K. If (gK)x=y and a neighbourhood Y of y is given, there is a neighbourhood U of g and X of x such that g^{'} \in X, x^{'} \in X imply g^{'}(x^{'}) \in Y. But then (g^{'}(K)(x^{'}) \in Y. From this it is clear that T^{*} is continuous simultaneously in gK and x. It can be seen that

g_{1}K(g_{2}K(x)) = g_{1}g_{2}K(x)

It follows also that gK(x)=x for every x implies g \in K, equivalently gK=K.

1.28.1

If G is a transformation group of M and h is a homeomorphism of M onto itself, then the homeomorphisms:

\{ hgh^{-1}\}

of M onto itself form a transformation group which is said to be topologically equivalent to G.

1.26.2

A topological group G can be regarded as a transformation group on itself as space in several ways, in particular by associating with a \in G.

  1. a(g) = ag (left translation)
  2. a(g) = ga^{-1} (right translation)
  3. a(g) = aga^{-1} (conjugation, taking of transforms.
    Also G is a transformation group of a left coset space G/H where H is a closed subgroup, by
  4. a(gH) = agH In cases (1) and (2) G is effective. In case (4) if agH = gH for every g \in G then a \in gHg^{-1} for every g \in G. Then K = \bigcap gHg^{-1} is an invariant subgroup of G, and G/K is effective on G/H. Of course, K is a subgroup of H which depends on H as well as G and which may be trivial.

1.26.3

Further examples of transformation groups are given below, proofs are omitted. (PS: I will supply the proofs in a later blog, most probably after this blog):

  1. Let G = GL(n,R) be the real n x n matrices where a = (a_{ij}) with |a_{ij}| neq 0. Let E_{n} be the space of n real variables u_{1}, u_{2}, \ldots, u_{n}. Then G is a transformation group of E_{n} whose elements are T_{a}^{*}: a(u) = u^{'} where u_{i}^{'} = \Sigma a_{ij}u_{j}
  2. With G as above, let S^{n-1} in E_{n} be the (n-1)-sphere defined by \Sigma u _{i}^{2}=1. Let G act on S^{n-1} as follows: T_{a}^{''}: a(u)=u^{''} where u_{i}^{''}=u_{i}^{'}/(\Sigma u_{i}^{2})^{1/2} and u_{i}^{'} is as above. Here the effective group is G/Kn where Kn consists of scalar matrices: (h\delta_{ij}) where \delta_{ij} is the Kronecker delta and h is positive.
  3. Let G be the group of two by two real matrices with determinant one and let it act on E_{2} as in (1).
  4. Let G be the group of two by two real matrices of determinant one and let G act on itself by inner automorphisms. As a space G is the product of a circle and a plane. One parameter groups fill a neighbourhood of the identity and in the large are closed sets which are either circles or lines. No two of them cross and they are permuted by G.
  5. For fixed integers m, n let G be the circle group acting on E_{4} as follows: x_{1}^{1}=x_{1}\cos{2\pi m t} + x_{2}\sin{2\pi mt}; x_{2}^{1} = -x_{1}\sin{2\pi mt} + x_{2}\cos{2\pi mt} ; x_{3}^{1}=x_{3}\cos{2 \pi n t} + x_{4}\sin{2 \pi n t}; x_{4}^{1}=-x_{3}\sin{2\pi n t}+x_{4}\cos{2 \pi nt}. This can also be viewed as a transformation group on the unit sphere S^{3} in E^{4} since it leaves S^{3} invariant. It is known that the simple closed curves swept out by points of S^{3} are linked.
  6. A quasi-relation in E_{3} in a fixed cylindrical coordinate system (z,r, \theta) is a group of homeomorphisms depending on a positive continuous function F(r,z), where 0 < r, < \infty, -\infty < z < \infty, F bounded on compact sets in E_{3} and given by (*) h_{t}: (z,r,\theta) \rightarrow (z,r, \theta + 2\pi F (r,z)t) for all real t.

Each point which is not a fixed point moves in a circle about the z-axis. The period of a moving point, that is the least positive t for which it is left fixed by (*) above varies continuously. If h is an arbitrary homeomorphism of E_{3} upon itself, then \{ h^{-1}h_{1}h\} defines a topological quasi-rotation group. As we shall mention later, these groups can be characterized abstractly.

1.26.4

The transformation group G is called transitive on M if for every x, y \in M there is at least one g \in G such that g(x)=y. As remarked earlier every topological group G is transitive on G/H. where H is a closed subgroup.

THEOREM.

Let (G,M) be a topological transformation group which is transitive on M. Then the groups of stability G_{x} where x \in M are conjugate and for any one of them G/G_{x} is mapped in a continuous one-one way onto M by the map:

T_{1}: gG_{x} \rightarrow g(x)

Let x, y \in M be given with g(x)=y. If g^{'}(x)=x, gg^{'}g^{-1}(y)=y. It follows that G_{x} and G_{y} are conjugate.

Now let x be fixed. If g^{'}(x)=x, gg^{'}(x)=g(x) for every g \in G and this shows that the map

T_{1}: G/G_{x} \rightarrow M defined in the theorem is one-one. It maps G/G_{x} onto M because G is transitive.

Let T be the natural map G \rightarrow G/G_{x}. Then T_{1}T maps G onto M T_{1}Tg=g(x).

This map is continuous in G by the definition of a transformation group. Let U be open in M. Then (T_{1}T)^{-1}U is open in G and T(T_{1}T)^{-1}U is open in G/G_{x}, since F is an open map. This shows that T_{1} is continuous as well as one-one. In the cases of most interest it will turn out that T_{1} is a homeomorphism.

1.27 LOCALLY EUCLIDEAN SPACES

We have used E_{n}, where n \in \mathcal{N} to denote euclidean n-space, with real coordinates x_{1}, x_{2}, \ldots, x_{n}. It is the topological class of spaces homeomorphic to E_{n} which we have in mind, rather than the space endowed with a standard euclidean metric. By the Brouwer theorem, an open subset of E_{m} and an open subset of E_{n} cannot be homeomorphic if m \neq n. This shows that the possibility of the one-one bicontinuous coordinatization (x_{1}, x_{2}, \ldots, x_{n}) of E_{n} is a topological property. This number n is a topological invariant and is called the dimension.

The term locally euclidean is used to describe a topological space E of fixed dimension n each point of which has a neighbourhood that is homeomorphic to an open set in E_{n}. The simplest examples of such spaces are the open subsets of E_{n}. If a locally euclidean space is connected, it is called a MANIFOLD. For example, the spheres of all dimensions, the ordinary torus, the cylinder, etc. are manifolds.

A locally euclidean space can be covered by a certain number (not necessarily finite) of open sets each homeomorphic to an open set in E_{n}; let us call such sets each with a fixed homeomorphism, a coordinate neighbourhood. The circle C_{1} can be covered by two (or more) coordinate neighbourhoods, the two dimensional sphere S^{2} by two or more. However, to describe classical euclidean space one uses the entire family of those coordinate systems which are related to each other by orthogonal transformations. Similarly to define affine n-dimensional space, one uses a larger family, namely, all coordinate systems which are affinely related to each other.

We describe a topological manifold one could use in it the family of all coordinate neighbourhoods. However, if for some purpose a restricted class of coordinate neighbourhoods covering the manifold is specified, then one can speak of admissible coordinate systems. In general, where two coordinate neighbourhoods overlap, the coordinate systems will not be found to be in any simple relation. It may happen that the admissible coordinate have been so selected that in every region of overlap of two such systems the two sets of coordinates are related by functions which are differentiable or analytic.

A manifold is said to be a differentiable manifold and to have a differentiable structure of class C^{r} where r is greater than or equal to 1, if there is given a covering family of coordinate neighbourhoods in such a way that where any two of the neighbourhoods overlap the coordinate transformation in both directions is given by n functions with continuous, partial derivatives of order r. A manifold may have essentially different differentiable structures (Milnor). A manifold need not possess a differentiable structure (Kervaire, Smale). The n-sphere (with the possibe exception of n=3) has only a finite number of such structures (see Kervaire-Milnor).

In the same way a manifold is said to be a real analytic manifold and to have a real analytic structure if there is given a covering family of coordinate neighbourhoods in such a way that where any two overlap the coordinate transformation in both directions is given by n functions which are real analytic, that is in some neighbourhood of each point of the overlap they can be expanded in power series.

The definition of a complex analytic manifold and structure is similar to the above. Such a manifold of course has an even number of real dimensions; it is automatically real analytic. However, there are many real analytic manifolds of even dimension which cannot be given a complex analytic structure; thus the existence of a complex analytic structure is a much stronger property than the existence of a differentiable or even real analytic structure.

1.27.1

Suppose that M is an n-dimensional manifold and that x and y are points of M belonging to a set U which is homeomorphic to an open n-sphere. Then it is not difficult to describe a homeomorphism of M onto itself which keeps fixed all points of M not inside of U, and which maps x onto y. Using this and using the connectedness of M, and being given an arbitrary pair of points x and y of M one can find a homeomorphism of M onto itself mapping x on y. The details are not presented here.

1.27.2

Suppose that M and N are manifolds and that there is given a continuous map f of M onto M^{'}. The map f is called a covering map and M is said to cover M^{'} if the following conditions are satisfied:

a) for each y in M^{'} there is an open neighbourhood V of y such that f^{-1}(V) is the union of disjoined open sets U_{x} where there is a U_{x} for each x \in f^{-1}(y) and x \in U_{x}

b) f is a homeomorphism of U_{x} onto V for each x in f^{-1}(y). For each y \in M^{'}, each x \in f^{-1}(y) is called a covering point. It is clear that M and M^{'} are of the same dimension, and it is clear that each point of f^{-1}(y) is an isolated point of f^{-1}(y) (each point is a relatively open subset) so that f^{-1}(y) is a discrete set.

By way of example, let M^{'} be the ordinary torus with momentarily convenient coordinates u, v: 0 \leq u, v <1 and let M be the ordinary plane with real coordinates x and y. Define the map f(M) = M^{'} by

(x,y) \rightarrow (u,v) if and only if x \equiv u and y \equiv v {\pmod 1} This pair of manifolds can also be regarded as an example of a group M covering a factor-group M^{'}. Thus let M now denote the two-dimensional vector space V_{2} and let x_{1} and y_{1} denote two independent vectors in V_{2}. Let D be the countable, discrete subgroup of V_{2} consisting of the linear combinations of x_{1} and y_{1} with integral coefficients. Finally, let M^{'} now denote the toral group V_{2}/D.

A more general example is the following:

If G is any connected locally euclidean group and H is a discrete subgroup of G then G covers the coset-space G/H under the natural map G \rightarrow G/H.

A manifold M^{'} is called simply connected if whenever it is covered by a manifold M, the covering map f: f(M) = M^{'} is a homeomorphism (in that case f^{-1}(y) is single-valued). Euclidean spaces of all dimensions and the sphere-spaces of dimension greater than one are simply connected manifolds; the one-dimensional sphere (circumference of a circle) and more generally the toral spaces of all dimensions are not simply connected.

Cheers,

Nalin Pithwa

Topological Spaces and Groups: Part 2: Fast Review

Reference: Chapter 1, Topological Transformation Groups by Deane Montgomerry and Leo Zippin

1.10 Sequential Convergence:

The proof of the principal theorem of this section illustrates a standard technique. It will involve choosing an infinite sequence, then an infinite subsequence, then again an infinite subsequence and so on repeating this construction a countably infinite number of times. A special form of this method is called the Cantor diagonalization procedure. To facilitate the working of this technique we shall sometimes use the following notation:

1.10.1

The letter I will denote the sequence of natural numbers 1, 2, 3, …When subsequences of I need to be chosen, they will be labelled in some systematic way: I_{1}, I_{2}, \ldots or I^{'}, I^{''}, \ldots or I^{*}, I^{**}, \ldots and so on. Then given a sequence of elements x_{n}, n \in I, we can refer to a subsequence as : x_{n}, n \in I_{1}, or : x_{n}, n \in I^{*} and so on.

DEFINITION:

Let S denote a metric space and let K_{n}, n \in I be a sequence of subsets of S. The sequence K_{n} is said to converge to a set K if for every \epsilon >0

1.10.3 K_{n} \subset S_{\epsilon}(K) and K \subset S_{\epsilon}(K_{n})

for n sufficiently large (depending only on \epsilon)

If K_{n} is given and if a K exists satisfying relation 1.10.3 then \overline{K} also satisfies 1.10.3. If two closed sets K^{'} and K^{''} satisfy 1.10.3 for the same sequence K_{n} then K^{'}=K^{''}. In the special case that the sets K_{n} are single points, the set K if it exists is a point and is unique.

1.10.4 THEOREM

Every sequence of non-empty subsets of a compact metric space S has a convergent subsequence.

Proof:

Let K_{n}, n \in I be an arbitrary sequence of subsets of S and let W_{n}, n\in I be a basis for open sets in S. (Theorem 1.9)

Let I be called I_{0} and suppose a sequence I_{m-1} has been defined. Consider W_{m} \bigcap K_{n}, n \in I_{m-1} m fixed. Then either W_{m} \bigcap K_{n} is not empty for an infinite subsequence of integers n \in I_{m-1} or on the contrary W_{m} \bigcap K_{n} is empty for almost all n \in I_{n-1}. In the first of these cases define I_{m} as the set of indices n in I_{m-1}. In the first of these cases define I_{m} as the set of indices n in I_{m-1} such that W_{m} \bigcap K_{n} is not empty for n \in I_{m}. Then in all possible cases I_{m} \subset I_{m-1} is uniquely defined. We now consider I_{m} to be defined by induction for all m \in I.

We can now specify what subsequence of K_{n} we may take as convergent. Let I^{*} \subset I denote the diagonal sequence of the sequences I_{m}, that is I^{*} contains the m-th element of I_{m} for each m. We shall show that K_{n}, n \in I^{*} is convergent. It follows from the definition of I^{*} that for each m \in I, if we except at most the first m integers in I^{*},

W_{m} \bigcap K_{n}, n \in I^{*},

is always empty or is never empty depending on m.

We next define the set K to which the sets K_{n}, n \in I^{*} will be shown to converge. Let W denote the union of those sets W_{m}, m \in I for which W_{m} \bigcap K_{n}, n \in I^{*} is almost empty. Let K = S - W Since each W_{m} forming W meets at most a finite number of the sets K_{n} no finite number of these W_{m} can cover S. Therefore W cannot cover S, since S is compact, and hence K is not empty. The set K is closed and therefore compact.

Let \epsilon >0 be given. There is a covering of K by sets W_{k_{1}}, W_{k_{2}},  \ldots, W_{k_{s}}, k_{i} \in I

each meeting K and each of diameter less than \epsilon. None of the sets W_{k_{i}} can belong to W and each must intersect almost all the K_{n}, n \in I^{*}. Therefore for sufficiently large n, n \in I^{*}

W_{k_{i}} \bigcap K_{n} \neq \Phi

It follows that K \subset S_{\epsilon}(K_{n}), n \in I^{*} for all n sufficiently large. Finally it can be seen that the closed set S - S_{\epsilon}(K) \subset W. It follows that the complement of S_{\epsilon}(K) is covered by sets W_{j_{1}}, W_{j_{2}}, \ldots, W_{j_{t}} each of which is an element in the union defining W. Therefore W_{j_{k}} \bigcap K_{n}, n \in I^{*} k=1,2, \ldots t

is almost always empty. Therefore for sufficiently large n \in I^{*}

K_{n} \subset S_{\epsilon}(K).

This completes the proof of the theorem. QED.

1.10.5

Let X and Y be closed subsets of a compact metric space S. Define Hausdorff metric: d(X,Y) as the greatest lower bound of all \epsilon such that symmetrically X \subset S_{\epsilon}(Y), Y \subset S_{\epsilon(Y)}

This is a metric for the collection of closed subsets of S.

THEOREM:

The set F of all closed subsets of a compact metric space S is a compact metric space in the metric defined above.

Proof.

The set F is a metric space in the metric defined above. It A_{n} is in F then A_{n} has a subsequence converging to a set A in the sense of convergence defined above and the set A may be assumed closed. It follows that the subsequence also converges to A in the sense of the metric of F. Hence, every sequence in F has a convergent subsequence.

Let W_{n} where n \in \mathcal{N} be a basis for open sets in S as in the preceding Theorem. For each n and each choice of integers k_{1}, k_{2}, \ldots, k_{n} let W(k_{1}, k_{2}, \ldots, k_{n}) denote the union of the sets W_{k_{1}}, W_{k_{2}}, \ldots, W_{k_{n}}. Now in F let W^{*}(k_{1}, \ldots, W_{k_{n}}) consist of all the compact sets in S which belong to W(k_{1}, \ldots, k_{n}) and meet each W_{k_{i}}. This gives a countable collection of subsets of F. The proof of the preceding theorem shows that this collection is a basis for open sets in F.

Finally, let \{ O_{n}\} where n=1, 2, …be a countable collection of open sets of F which cover F. We have to find an integer m such that \bigcup_{1}^{m}O_{i} covers F. If no such integer existed we could find a sequence of points x_{n} where x_{n} \in F - \bigcup_{1}^{n}U_{i}. This sequence would have to have a susbsequence converging to some point x. Since x \in O_{m} for some m, it follows that infinitely many of the x_{n} belong to O_{m}; this contradiction proves the theorem.

QED.

1.10.6

Note that separability implies that every collection of covering sets has a countable covering subcollection. It also implies that there exists a countable set of points which is everywhere dense in the space.

THEOREM.

A metric space S is compact if and only if every infinite sequence of points has a convergent subsequence.

Proof:

Suppose that every infinite subsequence of points of S has a convergent subsequence. We shall prove that S is separable. The last paragraph of the preceding section then shows that S is compact. The converse is shown in 1.10.4

For each positive integer n construct a set P_{n} such that (1) every point of P_{n} is at a distance at least \frac{1}{n} from every other point of P_{n} (2) every point of S not in P_{n} is at a distance less than 1/n from some point of P_{n}. It is easy to see that no sequence of points in any one P_{n} can be convergent and it follows that P_{n} is a finite point set. Let P= \bigcup P_{n}. Then P is countable and every point of S is a limit point of P. For each rational r >0 and each point of P construct the “sphere” with that point as centre and radius r. The set of these spheres is countable and is a basis for open sets. This concludes the proof.

QED.

EXAMPLE.

Let S be a compact metric space let H be the space of real continuous functions defined on S with values in R_{1}. Each continuous function f(x) determines a closed subset of S \times R_{1} namely the graph consisting of the pairs (x, f(x)), x \in S. Hence H is a subset of a compact metric space (see examples 2 and 3 in Sec 1.9).

Example 2 of Section 1.9 reproduced below:

The set F of continuous functions defined on a compact space S with values in a metric space M becomes a metric space by defining for f, g in F d(f,g) = lub (x \in S) [d_{M}(f(x), g(x))] where d_{M} is the metric in M. Recall also the following here in this example: Theorem: Let S be a compact space and \{ D_{a}\} a collection of closed subsets such that \bigcap_{a}D_{a} is empty. Then there is some finite set D_{a_{1}}, \ldots, D_{a_{n}} such that \bigcap_{i}D_{a_{i}} is empty. From this theorem it follows that: A lower semi-continuous (upper semi-continuous) real valued function on a compact space has finite g.l.b. (and L.u.b) and always attains these bounds at some points of space.

Example 3 of Section 1.9 reproduced below:

If S is a compact metric space and E_{1} denotes the real line, then S \times E_{1} is a metrizable locally compact space with a countable basis for open sets.

Remarks: Notice that the metric defined for H in example 2 of 1.9 and the metric which it gets from S \times H_{1} are topologically equivalent. This proves that H itself is a separable metric space.

QED.

1.11 TOPOLOGICAL GROUPS

Topological groups were first considered by Lie, who was concerned with groups defined by analytic relations (to be discussed later). Around 1900-1910 Hilbert and others were interested in more general topological groups. Brouwer showed that the Cantor middle third set can be made into an abelian topological group. Later Schreier and Leja gave a definition in terms of topological spaces whose theory had been developed in the intervening time.

A topological group is a topological space whose points are elements of an abstract group, the operations of the group being continuous in the topology of the space. A detailed definition containing some redundancies is as follows:

DEFINITION:

A topological group G is a space in which for x, y \in G there is a unique product xy \in G, and

i) there is a unique identity element e in G such that xe = ex =x for all x \in G

ii) to each x \in G there is an inverse x^{-1} \in G such that xx^{-1} = x^{-1}x=e

iii) x(yz) = (xy)z for x, y, z \in G

iv) the function x^{-1} is continuous on G and xy is continuous on G \times G

Familiar examples are the real or complex numbers under addition with the usual topologies for E_{1} and E_{2} respectively, and the complex numbers of absolute value one under multiplication with their usual topology as a subset of E_{2}. The space of this last group is homeomorphic to the circumference of a circle.

1.11.1

Of course, properties 1, 2 and 3 define a group in the customary sense and a topological group may be thought of as a set of elements which is both an abstract group and a space, the two concepts being united through 4. Wnen a subset H of G is itself a group we shall call H a subgroup of G, but we shall understand that H is to be given the relative topology. It is easy to see that then H becomes a topological group.

If H is a subgroup of G and if x and y are points of G belonging to the closure of H, then every neighbourhood of the product xy contains points of H. For, let U be a neighbourhood of xy. Then by property 4, there exists neighbourhoods V of x and W of y such that every product of an element of V and an element of W is contained in U. We see from this that xy belongs to the closure of H. Similarly, x^{-1} belongs to the closure. Thus, \overline{H} is a group, and we shall call it a closed subgroup.

1.11.2

If G_{a} where a \in \{ a\} is a collection of topological groups and if G denotes the product space defined in 1.6, (previous blog), then G can be regarded as a group (the product of two elements of G being defined by the product of their components in each factor G_{a}). Because each neighbourhood of the PROD G_{a} depends on only a finite number of the factors it is easy to see that G becomes a topological group. We shall call it the topological group of the factors G_{a}.

By way of examples, note that the product of an arbitrary number of groups each isomorphic to C_{1}: the group of reals modulo one (isomorphic to the complex numbers of modulus one under multiplication) is a compact topological group. The product of an arbitrary number of factors each isomorphic to the group of reals is a topological group which is locally compact if the number of factors is finite.

A finite group with the discrete topology is compact and the topological product of any collection of finite groups is therefore compact.

1.11.3

If x and y are in topological group, x \neq y, then as will be seen (section 1.16) we may choose a neighbourhood W of e such that

y \notin WW^{-1}

Hence, yW and xW are disjoined, and thus two distinct points of a topological group are in disjoined open sets. This is called the Hausdorff property (see section 1.10); it implies that a point is a closed set.

It is easy to see that if H is an abelian subgroup of a topological group G then \overline{H} is also abelian. Thus, if x, y \in \overline{H} and xy \neq yx there are neighbourhoods U_{1} of xy and U_{2} of yx with U_{1} \bigcap U_{2} = \Phi. There exist neighbourhoods V of x and W of y such that for every v \in V and w \in W vw \in U_{1} and wv \in U_{2}. However, if v, w \in H then wv = vw and we are led to a contradiction proving that the closure of H is abelian.

QED.

1.11.4

Important examples of topological groups are given below:

EXAMPLE 1.

The sets M_{n}(R) and M_{n}(C) of all n \times n matrices of real and complex elements under addition with the distance of A = (a_{ij}), B = (b_{ij}) defined by

d(A, B) = max_{i,j}|a_{ij} - b_{ij}|

The spaces of these two groups are homeomorphic to E_{n^{2}} and E_{2n^{2}}. They are in fact the sets of real or complex vectors with n^{2} co-ordinates, and hence are vector spaces as well as groups. Another example is the set H of continuous real valued functions on a metric space under addition.

EXAMPLE 2.

The sets of non-singular real or complex n \times n matrices GL(n,R), GL(n,C) under multiplication; these are subsets of M_{n}(R) and M_{n}(C) respectively and are given the induced topology. They are open subsets and are therefore locally compact and locally euclidean. (see 1.27 later)

EXAMPLE 3.

Let S be a compact metric space and let G be the group of all homeomorphisms of S onto itself topologized as a subspace of the space of continuous maps of S into itself. (Section 1.9 previous blog, example 2)

1.12 ISOMORPHISM of TOPOLOGICAL GROUPS

The spaces associated with two topological groups may be homeomorphic but the groups essentially different; for example, one abelian and the other not.

EXAMPLE 1.

The matrices \left |\begin{array}{cc}a & 0 \\ 0 & b \end{array}\right | where a and b are real numbers under addition. This is an abelian group with E_{2} as space.

EXAMPLE 2.

The matrices \left| \begin{array}{cc} e^{a} & b \\ 0 & e^{-a} \end{array} \right | where a and b are real under multiplication. This is a non-abelian group with E_{2} as space.

If we give the space in this example (or example 1) the discrete topology we obtain a new topological group with the same algebraic structure.

EXAMPLE 3.

In the additive group of integers, for each pair of integers h and k, k \neq 0 let the set \{ h \pm nk\} where n=0, 1,2, …, be called an open set and let the collection of all these sets be taken as a basis for open sets.

EXAMPLE 4.

Introduce a metric into the additive gorup of integers, depending on the prime number p, defined thus:

d(a,b) = \frac{1}{p^{n}}

if a \neq b and p^{n} is the highest power of p which is a factor of a-b.

EXAMPLE 5.

Let G be the integers under addition with any set called open if it is the complement of a finite set (or is the whole space or the null set). Algebraically G is a group and it is also a space. However, it is not a topological group because addition is now not simultaneously continuous. It is true however that addition is continuous in each variable separately.

For some types of group spaces separate continuity implies simultaneous continuity. It is not known whether this is true for a compact Hausdorff group space.

DEFINITION.

Two topological groups will be called isomorphic if there is a one-one correspondence between their elements which is a group isomorphism (preserves products and inverses) and a space homeomorphism (preserves open sets).

An isomorphic map of G onto G is called an automorphism.

In examples (3) and (4) the abstract group structure of the additve group of integers is embodied in infinitely many non-isomorphic topological groups.

1.13 SET PRODUCTS

If G is a group A \subset G let A^{-1} denote the inverse set \{ a^{-1}\} where a \in A. Clearly, (A^{-1})^{-1}=A. If B \subset G let AB denote the set \{ ab\} where a \in A, b \in B. It is understood that the product set is empty if either factor is empty. We shall write AA = A^{2} and so on. It can be seen that (AB)C = A (BC) and (AB)^{-1} = B^{-1}A^{-1}. The set AA^{-1} satisfies

(AA^{-1})^{-1}=AA^{-1}

that is, it is symmetric. Similarly, the set intersection A \bigcap A^{-1} is symmetric. The intersection of symmetric sets is symmetric.

A set H in G is called invariant if gH = Hg for every g \in G equivalently if gHg^{-1}=H.

THEOREM 1.

Let G be a topological group and let A \subset G be an open set. Then A^{-1} is open.

Proof:

Let a^{-1} be in A. By the continuity of the inverse there exists an open set B containing a such that b \in B implies b^{-1} \in A. This means that B^{-1} \subset A and therefore B \subset A^{-1}. Thus A^{-1} is a union of open sets and is open.

COROLLARY:

The map x \rightarrow x^{-1} is a homeomorphism.

LEMMA.

Let G be a topological group, A an open subset, b an element. Then Ab and bA are open.

Proof:

Let a \in A and let c=ab. Then a = cb^{-1}. Because “a” regarded as a product is continuous in c there must exist an open set C containing c such that if c^{'} \in C then c^{'}b^{-1} \in A. But then c^{'} \in Ab and it follows that C \subset Ab. Therefore Ab is a union of open sets and is open. The proof that bA is open is similar.

QED.

COROLLARY.

For each a \in G, the left and right translations : a \rightarrow ax and x \rightarrow xa are homeomorphisms.

THEOREM 2.

Let G be a topological group and let A and B be subsets. If A or B is open then AB is open.

Proof.

Since AB is a union of sets of the form Ab, b \in B, it is open if A is open. SimilarlyAB is a union of sets aB, a \in A and is open if B is open.

COROLLARY.

Let A be a closed subset of a topological group. Then Ab and bA are closed.

Proof:

This is true because left and right translations by the constant b are homeomorphisms of G onto G.

Now, a function f(x) taking a group G_{1} into another group G_{2} will be called a homomorphism if

*) f(x)f(y) = f(xy) where x, y \in G_{1}

When G_{1} and G_{2} are topological we shall ordinarily require that f be continuous. The most useful case is where f is open as well as continuous. In many situations (see 1.26.4 and 2.13 in later blogs) continuity implies openness but this is not true in general. The set of elements going into e is a subgroup and if f is continuous it is a closed subgroup (a point in a topological group is a closed set). This is the kernel of the homomorphism.

EXAMPLE.

Let V_{1} be the additive group of real numbers and G be a topological group. A continuous homomorphism h(t) of V_{1} into G is called a one-parameter group in G. If h is defined only on an open interval around zero satisyfying the definition of homomorphism so far as it has meaning, then h(t) is called a local one-parameter group in G. If h(t) is a one-parameter group, the image of V_{1} may consist of e alone and then h(t) is a trivial one-parameter group. If this is not the case and if for some t_{1} \neq 0 and h(t_{1})=e, then the image of V_{1} is homeomorphic to a circumference. In case h(t)=e only for t =0 the image of V_{1} is a one-one image of the line which may be homeomorphism of the line or a very complicated imbedding of the line. To illustrate this let G be a torus which we obtain from the plane vector group V_{2} by reducing mod one in both the x and y directions. In V_{2} any line through the origin is a subgroup isomorphic to V_{1} and after reduction the line y=ax is mapped onto the torus G thus giving a one-parameter group in G. If a is rational the image is a simple closed curve but if a is irrational the image is everywhere dense on the torus.

1.14 PRODUCTS of Closed Sets

If A and B are subgroups of a group G, AB is not necessarily a subgroup. However if A is an invariant subgroup (that is, g^{-1}Ag = A where g \in G) and B is a subgroup then AB is a subgroup.

The product of closed subsets, even if they are subgroups, need not be closed. As an example let G be the additive group of real numbers, H_{1} the subgroup of integers \{ \pm n\} and H_{2} the subgroup \{ \pm n\sqrt{2}\}. The product H_{1}H_{2} is countable and a subgroup but it is not a closed set.

It will be shown later that if A is a compact invariant subgroup and B is a closed subgroup then AB is a closed subgroup (corollary of 2.1 in later blogs). (Remark: I think in I N Herstein’s language of Topics in Algebra, an “invariant subgroup” is a normal subgroup. Kindly correct me if I am mistaken).

1.15 Neighbourhoods of the identity

Let G be a topological group and U an open subset containing the identity e. We showed in 1.13 that xU is open and clearly x \in xU. Conversely, if x \in O, O is open, then U = x^{-1}O is an open set containing e.

If a collection of open sets \{ U_{a}\} is a basis for open sets at e then every open set of G is a union of open sets of the form x_{a}U_{a}, where x_{a} \in G, U_{a} \in \{ U_{a}\} and the topology of G is completely determined by the basis at e. In particular the collection \{ xU_{a}\} is a basis for open sets at x so also is \{ U_{a}x\}

If U is a neighbourhood of e, U^{-1} is a neighbourhood of e and U \bigcap U^{-1} is a symmetric neighbourhood of e.

THEOREM

Let G be a topological group and U a neighbourhood of e. There exists a symmetric neighbourhood W of e such that W^{2} \subset U.

Proof:

Since e.e = e and the product is simultaneously continuous in x and y there must exist neighbourhoods V_{1} and V_{2} of e such that V_{1}V_{2} \subset U. Define W = V_{1} \bigcap V_{1}^{-1} \bigcap V_{2} \bigcap V_{2}^{-1}. Then W^{2} \subset U and this completes the proof.

QED.

COROLLARY.

Let G be a topological group. If x \neq e there exists a neighbourhood W of e such that W \bigcap xW is empty.

Proof.

Since G is a topological space there is a neighbourhood V_{1} of e not containing x or there is a neighbourhood V_{2} of e not containing x. In either case there is a neighbourhood of e not containing x. Let W be a symmetric neighbourhood of e, W^{2} \subset U. If W \bigcap xW were not vacuous there would exist w, w^{'} \in W with w^{'}=w. But this gives x = w^{'}w^{-1} \in W^{2} \subset U which is false.

QED.

1.15.1

If G is a topological group then a set S of open neighbourhoods \{ V\} which forms a basis at the identity e has the following properties:

(a) the intersection of all V in S is \{ e\}

(b) the intersection of two sets of S contains a third set of S

(c) given U in S there is a V in S such that VV^{-1} \subset U

(d) if U is in S and a \in U then there is a V in S such that Va \subset U

(e) If U is in S and a is in G there is a V in S such that aVa^{-1} \subset U.

Conversely, a system of subsets of an abstract group having these properties may be used to determine a topology in G as will be formulated in the following theorem, the proof of which is contained in main part in remarks already made.

THEOREM.

Let G be an abstract group in which there is given a system S of subsets satisfying (a) to (e) above. If open sets in G are defined as unions of sets of the form Va, a \in G then G becomes topological with S a basis for open sets at e. This is the only topology making G a topological group with S a basis at e.

1.16 COSET Spaces

Let G be a group and H a subgroup. The sets xH and yH where x, y \in G either coincide or are mutually exclusive; and xH = yH if and only if x^{-1}y \in H. Each set xH is called a coset of H, more specifically a left coset. Right cosets Hx, Hy will be used infrequently. We use the notation G/H for the set of all left cosets. When G is a (topological space) G/H will be made into a space (see below 1.16.2) but we speak of it as a space in the present case also, although it carries no topology at present.

If H is an invariant subgroup, that is if x^{-1}Hx=H equivalently if xH=Hx for any x \in G, then

xHyH = xyH where x, y \in G

is a true equation in sets of elements of G. In this case, the coset space becomes a group, the factor group G/H.

1.16.1

DEFINITION

By the natural map T of a group G onto the coset space G/H, H being a subgroup of G, we mean the map

T : x \rightarrow xH, where x \in G, xH \in G/H

For any subset U \subset G we have

T^{-1}(T(U)) = UH \subset G

Let G be a topological group, H a subgroup. It is useful to topologize the coset space and to do this so that the natural map T is continuous. It will become clear as we proceed that unless the group H is a closed subgroup of G, it will not be possible in general to have T continuous and G/H a topological space; for this reason only the case where H is closed will be considered.

1.16.2

DEFINITION.

Let G be a topological group and let H be a closed subgroup of G, that is a subgroup which is a closed set. By an open set in G/H we mean a set whose inverse under the natural map I is an open set in G/H.

THEOREM.

With open sets defined as above, G/H is a topological space and the map T is continuous and open. If xH \neq yH, there exist neighbourhoods W_{1} and W_{2} of xH and yH respectively such that W_{1} \bigcap W_{2} is empty.

Proof:

Let U denote an arbitrary open set in G. Then U/H is open (1.13) and is the inverse of T(U). Since T(U) is open in G/H, T is an open map.

Let x, y \in G and xH \neq yH. Then x \in yH and yH is closed because H is closed. There exists a neighbourhood U of e such that Ux \bigcap yH is empty. Let W be open, e \in W and W^{*} \subset U (Theorem in 1.15 above). Then if $latex WxH \bigcap WyH is not empty we can find w, w_{1} \in W and h, h_{1} \in H so that w_{1}xh = wy h_{1}. But this leads to w^{-1}w_{1}x = yh_{1}h^{-1} and implies that Ux meets yH. Therefore WxH \bigcap WyH is empty. The sets Wx and Wy are open. Therefore W_{1}=T(Wx) and W_{2} = T(Wy) are open in G/H; xH \in W_{1} yH \in W_{2} and W_{1} \bigcap W_{2} is empty. This is the main part of the proof and depends on the fact that H is closed. We have proved more than condition (4) of 1.1 (The T_{0} separation axiom) The remaining three conditions of 1.1 are easy to verify. The fact that T is a continuous map is stated in the definition of open set in G/H. The fact that T is open was proved in the last paragraph.

QED.

COROLLARY.

If G is a topological group, x, y \in G where x \neq y, then there exist neighbourhoods W_{1} of x and W_{2} of y such that W_{1} \bigcap W_{2} is empty.

Proof:

To see this it is only necessary to take H=e. A space in which every pair of distinct points belong to mutually exclusive open sets is called a Hausdorff space. Therefore it has been shown that a topological group G and a coset space G/H, H closed in G, are Hausdorff spaces.

COROLLARY.

Suppose that G is a topological group and H a closed invariant subgroup. Then with the customary definition of product : (xH)(yH) = xyH,, G/H becomes a topological group. The natural map of G onto G/H is a continuous and open homomorphism.

1.17 A FAMILY OF NEIGHBOURHOODS

Suppose that we are given a topological group G and a sequence of neighbourhoods of e: Q_{0}, Q_{1}, \ldots By repeated of the following Theorem 1.15: ( Let G be a topological group and U a neighbourhood of e. There exists a symmetric neighbourhood W of e such that W^{2} \subset U) .We can choose a sequence of symmetric open neighbourhoods of e : U_{0}, U_{1}, \ldots with U_{0}=Q_{0} such that

(1) U_{n+1}^{2} \subset U_{n}\bigcap Q_{n} where n=0, 1, ….

In this section we shall show how to imbed the sets U_{n} in a larger family of neighbourhoods possessing a multiplicative property which generalizes (1). We shall use this family in the next section to construct a real and non-constant function which is continuous on G. In 1.22 we shall use a similar family in order to construct a metric in a metrizable group. We remark in passing that in groups which do not satisfy the first countability axiom the set: \bigcap U_{n} may be of considerable interest (to be discussed in a future blog section 2.6); it is a closed group (if x, y \in \bigcap U_{n} then for every n, xy \in U_{n+1}^{2} \subset U_{n}).

Now, for each dyadic rational r = \frac{k}{2^{n}}, for n=0, 1, …and k=1, \ldots, 2^{n} we define an open neighbourhood V_{r} of e as follows:

(2) V_{1/2^{n}}=U_{n} for all n

and then using (3) and (4) alternately by induction on k.

(3) V_{2k/2^{n+1}}=V_{k/2^{n}}

(4) V_{(2k+1)/2^{n+1}} = V_{1/2^{n+1}}V_{k/2^{n}}

Each V_{n} depends on the dyadic rational r, and not on the particular representation by k/2^{n}. The entire family has the property:

(5) V_{1/2^{n}}V_{m/2^{n}} \subset V_{m+1/2^{n}} where m+1 \leq 2^{n}

For m=2k, (5) is an immediate consequence of (3) and (4). For m=2k+1 the left side of (5) becomes: V_{1/2^{n}}(V_{1/2^{n}}V_{k/2^{n-1}}) \subset V_{1/2^{n-1}}V_{k/2^{n-1}}

The right side of (5) becomes V_{(k+1)/2^{n-1}}. This sets up an induction on n and since (5) holds for n=1, we have proved that (5) is true for all n. It follows from (5) and also more directly that:

(6) V_{r} \subset V_{r^{'}} if r<r^{'} \leq 1

1.18 COMPLETE REGULARITY

THEOREM.

Suppose that G is a topological group and that F is a closed subset of G not containing e. Then one can define on G a continuous real function f, 0 \leq f(x) \leq 1, f(e) =0, f(x)=1 if x \in F.

Proof:

In virtue of the property described in the theorem, G is called a completely regular space at the point e. The theorem is due to Pontrjagin.

Set Q_{a}=G-F and setting Q_{n}=Q_{0} for every n, construct a family of sets V_{r} as in the preceding section.

Define f(x), x \in G as follows:

(1) f(x)=0 if x \in V_{r} for every r

(2) f(x)=1 if x \in V_{1}

and in all other cases

(3) f(x)=lub_{r} where \{ r \leq 1, x \in V_{r}\}

It is clear that e belongs to every V_{r} and that F = G - V_{1} and does not meet V_{1} so that there remains only to prove that f is continuous; let \epsilon>0 be given and let n be a positive integer such that 1/2^{n}<\epsilon.

Now suppose that f(x) <1 at some point x \in G. Then there is a pair of integers m and k such that k >n (same n as above), m < 2^{n} and (interpreting V_{0}, if it occurs, as the null set),

x \in V_{m/2^{k}}-V_{m-1/2^{k}}

Let y be an arbitrary element in the neighbourhood V_{1/2^{k}}x. Then,

y \in V_{1/2^{k}}V_{m/2^{k}} \subset V_{(m+1)/2^{k}}

By the choice of y, yx^{-1} \in V_{1/2^{k}}; therefore xy^{-1} \in V_{1/2^{k}} and x \in V_{1/2^{k}}y. It follows from this that y cannot belong to V_{(m-2)/2^{k}} and this shows that

(m-2)/2^{k} \leq f(y) \leq (m+1)2^{k}

Concerning x we know that

(m-1)/2^{2^{k}} \leq f(x) \leq m/2^{k} and we may conclude from both inequalities that

|f(x)-f(y)| \leq 2/2^{k} \leq 1/2^{n}<\epsilon

Suppose next that f(y)=1 and choose k>n as before. Let y be an arbitrary element in V_{1/2^{k}}x. Now y cannot belong to V_{m/2^{k}} with m < 2^{k}-2 without implying that f(x)<1. It follows that

1-2/2^{k} \leq f(y) \leq 1

and again we get |f(x) - f(y)| \leq e

This concludes the proof and f(x) is continuous on G.

QED.

COROLLARY.

A topological group is completely regular at every point.

1.19 HOMOGENEOUS SPACES

THEOREM.

Let G be a topological group, H a closed subgroup. Each element of G determines a homeomorphism of the coset space G/H onto itself, and G becomes a group of homeomorphisms of this space (topological space); furthermore G is transitive on the space: that is, each point may be carried to any other by an element of G.

Proof

Let a \in G. Associate to a the mapping T_{a}: xH \rightarrow axH.

This is a one-one transformation of G/H onto itself with T_{a^{-1}} as inverse. These transformations are open and each transformation is a homeomorphism.

Since T_{b}T_{a} is given by

T_{b}T_{a}(xH) = T_{b}(axH)=baxH = T_{ba}(xH).

the association of a \in G and T_{a} makes a group of transformations of G/H. It is clear that xH is carried to yH by T_{a} with a = yx^{-1}. This completes the proof.

QED.

A space is called HOMOGENEOUS when a group of homeomorphisms is transitive on it. We have shown that G/H is homogeneous with G being the transitive group of homeomorphisms. This implies that the unit segment R_{1}, for example, cannot be the underlying space of a group or even a coset space G/H since an end point of R_{1} cannot go to an interior point to a homeomorphism of R_{1}.

It follows from the simultaneous continuity of ax \in G in the factors a and x that the image point axH of xH under T_{a} is continuous simultaneously in the counter point xH and the element T_{a} of G. This makes G an instance of what is called a topological transformation group of a space M which will be defined below.

However, we shall be principally concerned with transformation groups which are locally compact and separable, acting on spaces which are topologically locally euclidean.

1.20 LOCAL GROUPS

An open neighbourhood of the identity of a topological group when it is regarded as a space in the relative topology has some of the properties of a group. There will usually be pairs of elements for which no product element exists in the neighbourhood. A structure of this kind is called a local group and will be defined below. Local groups often arise in a natural way, especially in the case of analytic group (Lie groups of transformations) and they have been intensively studied in that form

DEFINITION.

A space G is called a local group if a product xy is defined as an element in G for some pairs x, y in G and the following conditions are satisfied:

i) there is a unique element e in G such that ex and xe are defined for each x in G and ex = xe=x.

ii) If x, y are in G and xy exists then there is a neighbourhood U of x and a neighbourhood V of y such that if x^{'} \in U and y^{'} \in V then x^{'}y^{'} exists. The product xy is continuous wherever defined.

iii) The associative law holds whenever it has meaning.

iv) If ab=e then ba=e. An element b satisfying this relation is called an inverse and is denoted by a^{-1}. We assume that a^{-1} is unique and continuous where defined and that if it exists for an element a it exists for all elements in some neighbourhood of a. Note that a^{-1} always exists in some neighbourhood of e. In fact, there exists a symmetric open neighbourhood U of e such that U^{2} is defined.

The above definition is somewhat redundant.

EXAMPLE.

Any neighbourhood O of the identity oa topological group is a local group if the neighbourhood is open.

We shall call two local groups isomorphic if there is a homeomorphism between their elements which carries inverse to inverse and product to product in so far as they are defined. However, in some applications, it is natural to regard two local groups as equivalent if they belong to the same local equivalence class, that is, a neighbourhood of e in one is isomorphic to a neighbourhood of e in the other. In this book an isomorphism and preserves group operations so far as they are defined.

LEMMA.

Let G be a local group with U the symmetric open neighbourhood of e described in the definition. Given any neighbourhood V of e, V \subset U, there is a symmetric neighbourhood W of e, W^{3} \subset V. The product sets AB, BC, (AB)C, A(BC) exist for A, B, C \subset W and A(BC) = (AB)C. The set AB is open if either A or B is open. The sets A, bA, and Ab are homeomorphic for b \in W. Any two points of W have homeomorphic neighbourhoods.

The proof is omitted, the details being as in 1.13, 1.14 and 1.15.

Regards,

Nalin Pithwa