mod04lec27 - Continuous functions on metric spaces - Part 1

We specialize to the case of continuous functions on metric spaces. Two definitions of continuity familiar from calculus are shown to be equivalent to the open set definition of continuity for metric spaces: the first one is the so-called epsilon-delta definition, and the second one is the limit definition of continuity. Three new concepts: convergence of sequences, Hausdorff spaces and metrizability, are introduced. it is shown that for Hausdorff spaces, sequences converge to at most one point. Characterizations of the closure of a set and continuous functions are given in terms of convergent sequences for metrizable spaces.