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TOPOLOGICAL GROUPS the Purpose of These Notes Is to Give A TOPOLOGICAL GROUPS MATH 519 The purpose of these notes is to give a mostly self-contained topological background for the study of the representations of locally compact totally disconnected groups, as in [BZ] or [B, Chapter 4]. These notes have been adapted mostly from the material in the classical text [MZ, Chapters 1 and 2], and from [RV, Chapter 1]. An excellent resource for basic point-set topology is [M]. 1. Basic examples and properties A topological group G is a group which is also a topological space such that the multi- plication map (g; h) 7! gh from G × G to G, and the inverse map g 7! g−1 from G to G, are both continuous. Similarly, we can define topological rings and topological fields. Example 1. Any group given the discrete topology, or the indiscrete topology, is a topological group. Example 2. R under addition, and R× or C× under multiplication are topological groups. R and C are topological fields. Example 3. Let R be a topological ring. Then GL(n; R) is a topological group, n2 and Mn(R) is a topological ring, both given the subspace topology in R . If G is a topological group, and t 2 G, then the maps g 7! tg and g 7! gt are homeomorphisms, and the inverse map is a homeomorphism. Thus, if U ⊂ G, we have U is open () tU is open () Ut is open () U −1 is open: A topological space X is called homogeneous if given any two points x; y 2 X, there is a homeomorphism f : X ! X such that f(x) = y. A homogeneous space thus looks topologically the same near every point. Any topological group G is homogeneous, since given x; y 2 G, the map t 7! yx−1t is a homeomorphism from G to G which maps x to y. If X is a topological space, x 2 X, a neighborhood of x is a subset U of X such that x is contained in the interior of U. That is, U is not necessarily open, but there is an open set W ⊂ X containing x such that W ⊂ U. If G is a group, and S and T are subsets of G, we let ST and S−1 denote ST = fst j s 2 S; t 2 T g and S−1 = fs−1 j s 2 Sg: The subset S is called symmetric if S−1 = S. We will let 1 denote the identity element of a group unless otherwise stated. The following result, although innocent enough looking, will be the most often used in all of the results which follow. 1 2 MATH 519 Proposition 1.1. Let G be a topological group. Every neighborhood U of 1 contains an open symmetric neighborhood V of 1 such that VV ⊂ U. Proof. Let U 0 be the interior of U. Consider the multiplication map µ : U 0 × U 0 ! G. Since µ is continuous, then µ−1(U 0) is open and contains (1; 1). So, there are open sets V1;V2 ⊂ U such that (1; 1) 2 V1 ×V2, and V1V2 ⊂ U. If we let V3 = V1 \V2, then V3V3 ⊂ U −1 and V3 is an open neighborhood of 1. Finally, let V = V3 \ V3 , which is open, contains 1, is symmetric, and satisfies VV ⊂ U. Proposition 1.2. If G is a topological group, then every open subgroup of G is also closed. Proof. Let H be an open subgroup of G. Then any coset xH is also open. So, [ Y = xH x2GnH is also open. From elementary group theory, H = G n Y , and so H is closed. Proposition 1.3. If G is a topological group, and if K1 and K2 are compact subsets of G, then K1K2 is compact. Proof. The set K1 × K2 is compact in G × G, and multiplication is continuous. Since the continuous image of a compact set is compact, K1K2 is compact. If X is a topological space, and A is a subset of X, recall that the closure of A, denoted A, is the intersection of all closed subsets containing A. A necessary and sufficient condition for x to be an element of A is for every open neighborhood U of x, U \ A is nonempty, which may be seen as follows. If x 62 A, then there is a closed set F which contains A, but x 62 F . Then U = X n F is an open neighborhood of x such that U \ A = ?. Conversely, if U is an open neighborhood of x such that U \ A = ?, then X n U is a closed set containing A which does not contain x, so x 62 A. Proposition 1.4. If G is a topological group, and H is a subgroup of G, then the topo- logical closure of H, H, is a subgroup of G. Proof. Let g; h 2 H. Let U be an open neighborhood of the product gh. Let µ : G×G ! G denote the multiplication map, which is continuous, so µ−1(U) is open in G × G, and contains (g; h). So, there are open neighborhoods V1 of g and V2 of h such that V1 × V2 ⊂ −1 µ (U). Since g; h 2 H, then there are points x 2 V1 \ H 6= ? and y 2 V2 \ H 6= ?. Since x; y 2 H, we have xy 2 H, and since (x; y) 2 µ−1(U), then xy 2 U. Thus, xy 2 U \ H 6= ?, and since U was an arbitrary open neighborhood of gh, then we have gh 2 H. Now let ι : G ! G denote the inverse map, and let W be an open neighborhood of h−1. Then ι−1(W ) = W −1 is open and contains h, so there is a point z 2 H \W −1 6= ?. −1 −1 Then we have z 2 H \ W 6= ?, and as before this implies h 2 H. Remark. Note that in the last part of the proof of Proposition 1.4, we have shown that the closure of a symmetric neighborhood of 1 is again symmetric. Lemma 1.1. Let G be a topological group, F a closed subset of G, and K a compact subset of G, such that F \ K = ?. Then there is an open neighborhood V of 1 such that F \ VK = ? (and an open neighborhood V 0 of 1 such that F \ KV 0 = ?). TOPOLOGICAL GROUPS 3 Proof. Let x 2 K, so x 2 GnF , and GnF is open. So, (GnF )x−1 is an open neighborhood of 1. By Proposition 1.1, there is an open neighborhood Wx of 1 such that WxWx ⊂ −1 (G n F )x . Now, K ⊂ [x2K Wxx, and K is compact, so there exists a finite number of n points x1; : : : ; xn 2 K, such that K ⊂ [i=1Wixi, where we write Wi = Wxi . Now let n \ V = Wi: i=1 For any x 2 K, x 2 Wixi for some i. Now we have V x ⊂ Wix ⊂ WiWixi ⊂ G n F: In other words, F \ V x = ?. Since this is true for any x 2 K, we now have F \ VK = ?. Remark. Note that from Proposition 1.1, the neighborhood V in Lemma 1.1 may be taken to be symmetric. Proposition 1.5. Let G be a topological group, K a compact subset of G, and F a closed subset of G. Then FK and KF are closed subsets of G. Proof. If FK = G, we are done, so let y 2 G n FK. This means F \ yK−1 = ?. Since K is compact, yK−1 is compact. By Lemma 1.1, there is an open neighborhood V of 1 such that F \ V yK−1 = ?, or FK \ V y = ?. Since V y is an open neighborhood of y contained in G n FK, we have FK is closed. 2. Separation properties and functions A topological space X is said to be T1 if for any two distinct points x; y 2 X, there is an open set U in X such that x 2 U, but y 62 U. This is equivalent to one-point sets being closed. If G is a topological group, then G being T1 is equivalent to f1g being a closed set in G, by homogeneity. A topological space X is said to be Hausdorff (or T2) if given any two distinct points x; y 2 X, there are open sets U; V ⊂ X, x 2 U, y 2 V , such that U \ V = ?. Recall the following basic properties of Hausdorff spaces. Exercise 1. If X is a Hausdorff space, then every compact subset of X is closed. Exercise 2. Let X be a topological space, and let ∆ = f(x; x) j x 2 Xg ⊂ X × X be the diagonal in X × X. Then X is Hausdorff if and only if ∆ is closed in X × X. Of course, if X is T2, then X is T1, but the converse does not hold in general. If G is a topological group however, the converse is true, which we now show. Proposition 2.1. Let G be a T1 topological group. Then G is Hausdorff. Proof. Given distinct g; h 2 G, take an open set U containing 1, such that gh−1 62 U, which we may do since G is T1. Applying Proposition 1.1, let V be an open symmetric neighborhood containing 1, such that VV ⊂ U. Now, V g is open and contains g, and V h is open and contains h.
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