chapter 3 quotient spaces of topological spaces

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Chapter 3 25 Quotient Spaces of Topological Spaces 3.2 QUOTIENT GROUPS OF TOPOLOGICAL GROUPS Let Q be the quotient map of a topological group G onto the quotient group G/H, that is, the mapping which orders to each x G its equivalence class xH. The finest topology on G/H for which Q is continuous is known as the quotient topology; we shall denote it by Q. 3.2.1 If H is a subgroup of a topological group G, G/H is the space of left cosets equipped with the quotient topology, and Q : G G/H is the canonical mapping, then (i) Q is an open, continuous mapping. (ii) For each a G, the neighborhoods of Q(a) are precisely the sets Q(aV), where V is a neighborhood of the neutral element e of G. 3.2.2 Let N be a normal subgroup of a topological group G,let Q : G G/N be the canonical mapping, and equip G/N with the quotient group structure and the quotient topology. Then (i) G/N is a topological group. (ii) Q is an open, continuous mapping. (iii) For each a G, the neighborhoods of Q(a) for the quotient topology are precisely the sets Q(a)Q(V), where V is a neighborhood of the neutral element e of G. (iv) G/N is separated if and only if N is a closed subset of G. 3.2.3 If G is any topological group and N = { e }, then G/N is a separated topological group.
Chapter 3 26 Quotient Spaces of Topological Spaces 3.2.4 Let H be a subgroup of a topological group G and let G/H be the space of left cosets. Then (i) G/H is separated if and only if H is closed in G. (ii) G/H is discrete if and only if H is open in G. 3.2.5 If G is any topological group and C is the connected component of the neutral element e, then (i) C is a closed normal subgroup of G. (ii) For each a G the connected component of a is aC = Ca. (iii) The quotient topological group G/C is separated and totally disconnected (i.e., its connected components are all singletons). 3.2.6 Let G and H be topological groups, let f : G H be a continuous homomorphism of G onto H, and assume that G is compact and H is separated. Then (i) H is compact ; (ii) The kernel N of f is a closed normal subgroup of G, and (iii) The mapping xN f(x) is a bicontinuous isomorphism of G/N onto H. 3.2.7 Let G and H be topological groups, let f : G H be a continuous homomorphism of G onto H with kernel N, then the following conditions are equivalent: (i) f is an open mapping. (ii) For every neighborhood V of eG, f (V) is a neighborhood of eH ;
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chapter 3 quotient spaces of topological spaces

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