4 1. BASIC TOPOLOGICAL GROUP THEORY

P R O O F . Let F C X be a closed subset and let g e G with g ^ F. By applying

^ - 1 we can assume g = 1. Now VF = G \ F is a neighborhood of 1. As (7 x G —* G is

continuous we have an open neighborhood Uofl with UxU C W\ So f7 = £/nf7 - 1

is an open neighborhood of 1 with (7 = U~x and U2 C W.

The closure L7" C W. For if h € U then every neighborhood of /i meets U, so

ft[/ meets [7, say hb = a where a,b eU. Now ft = a 6 _ 1 e £/£7_1 =U2 CW.

Set V = G\U. Then £/ and F are disjoint, p G [7, and F C V. D

When G be a topological group and H is a subgroup, i / carries the subspace

topology unless we explicitly specify to the contrary. So the open sets in H are the

U D H where U is open in G. It is easy to check that H is a topological group.

COROLLARY 1.1.6. Let H be a closed subgroup of a topological group G. If G

is locally compact then H is locally compact. If G is compact then H is compact.

P R O O F . A closed subspace of a compact HausdorfT topological space is com-

pact. G is Hausdorff (Tychonoff separation condition T2) by Lemma 1.1.5. •

1.2. Subgroups , Quotient Groups, and Quotient Spaces

The coset space G/H carries the quotient topology: if TT : G — » G/H denotes

the projection, 7r(g) = gH, then a set U C G/H is open if and only if 7r_1(Lr) is

open in G. Also, if U is open in G so is the union UH = \Jheh Uh of open sets,

and then TT(U) is open in G/H because UH = 7r~1(ir(U)). So the map TT is both

continuous and open. The open subsets of G/H are the sets TT(U) — {gH | g G U}

where U runs over the open subsets of G.

If H is a normal subgroup of G then G/H inherits a group structure from G,

and the map {gH.g'H) \-+ (gH)(g/H)~1 = gg'~ H is continuous. So then G/H is

a topological group if and only if its points are closed. In this connection, note that

G/H is T\ if and only if H is closed in G. So we have

L E M M A 1.2.1. Let H be a closed normal subgroup of G. Then G/H is a topo-

logical group, the projection TT : G — G/H is a continuous homomorphism, and H

is the kernel of TT. (Note that TT : G —» G/H is an open map.)

The converse is the topological version of the standard isomorphism theorem:

P R O P O S I T I O N 1.2.2. Let f : G — L be a continuous homomorphism between

topological groups. Let H C G be the kernel of f and let M = f(G) C L, the image

of f. Then f factors through a continuous injective homomorphism f of G/H onto

M.

In general one cannot expect / to be a homeomorphism. For example, let L be

the multiplicative group of all 2 x 2 diagonal matrices d i a g j e ^ 1 ^ , e^^-^} with #, (j)

real. Let G be the additive group of real numbers. Choose an irrational number

(3 and define / : G — L by f(t) = diag{e vC ~ It , e^^-^}. Since /3 is irrational, the

image of / can't be a closed curve, but its closure must be a torus, so that closure

is all of L. Thus the image M = f(G) is dense in L. Evidently / is one to one.