On some generalizations of metric spaces

Abstract

Let N be the set of all positive integers, and let Psi:N to X be a sequence converging to Z in X. Let z_i = Psi(i) for every I in N, and let Z = {z, z_1, z_2, z_3, . . .}. We call Z a convergent sequence, and we denote by Z_n the convergent sequence {z, z_n, z_n+l, z_n+2, . . .}. In this dissertation the properties of convergent sequences are used to study certain aspects of the mapping space from one topological space to another. The mapping space from X to Y with the compact-open topology is denoted C(X,Y). The mapping space from X to Y with the topology whose subbasic open sets are of the form {f|f(Z)subset of U}, where Z is a convergent sequence in X and U is open in Y, is denoted C_cs (X,Y) and called the cs-open topology. A comparison is made between the properties of the compact-open topology and those of the cs-open topology. In 1966 E. Michael defined a class of topological spaces, called Aleph_0-spaces, which contains the separable metric spaces, and showed that if X and Y are both Aleph_0-spaces, then C(X, Y) is an Aleph_0-space. It is shown here that C_cs (X,Y) is also an Aleph_0-space whenever X and Y are both Aleph_0-spaces. A collection (Script P) of subsets of a topological space X is called a cs-pseudobase if , whenever z subset u with Z a convergent sequence and U open in X, then Z_n subset P subset U for some n in N and some P in (Script P). It is shown that a regular space is an Aleph_0-space if and only if it has a countable cs-pseudobase. A cs-sigma-space is defined to be a regular space with a sigma-locally finite cs-pseudobase. The class of cs-sigma-spaces contains all Aleph_0-spaces and all metric spaces. The properties of cs-sigma-spaces are compared to those of Aleph_0-spaces, and it is shown that every first countable and every locally compact cs-sigma-space is metrizable. The main result is that if X is an Aleph_0-space and Y is a cs-sigma-space, then both C(X,Y) and C_cs(X,Y) are cs-sigma-spaces. Thus , in particular, if X is a separable metric space and Y is a metric space, C(X,Y) and C_cs (X,Y) are cs-sigma-spaces.