chapter 3 quotient spaces of topological spaces

Chapter 3 27
Quotient Spaces of
Topological Spaces
(iii) The natural isomorphism of G/N onto H is bicontinuous.
3.2.8 Let H be a subgroup of a topological group G. (X, ) be a topological
space and let D be an upper semi-continuous decomposition of X such
that every member of D is a compact subset of X and let D have the
quotient topology Q. If X is (i) Hausdorff, (ii) regular, (iii) locally
compact or (iv) satisfies the second axiom of countability, then D also
has
3.2.9 If G is a metrizable topological group and if N is a closed normal
subgroup of G, then the quotient topological group G/N is metrizable.
3.2.10 Let X be a commutative lattice group, and let E be a subgroup of X that is
an order-convex sublattice. Then X/E is a lattice, and for x, y in X, we
have Q(x y) = Q(x) Q(y) , where Q is the projection of X onto X/E.
3.3 QUOTIENT SPACES OF TOPOLOGICAL LINEAR SPACES
Let (X, ) be a topological vector(or linear) space over the field K , let E
be a subspace of X , and let Q be the quotient map of X onto X/E, that is, the
mapping which orders to each x X its equivalence class ^ x = x + E. Obviously,
Q is linear. The topology on X/E is the quotient topology Q, that is, the finest
topology on X/E for which Q is continuous. Thus the open sets in X/E are the sets
Q(A) such that A + E is open in X; since G + M is open in X whenever G is ,
Q(G) is open in X/E for every open subset G of X; hence Q is an open map.