Some Useful Facts That Are Actually General Topology

This § just establishes some standard verbiage for things that are defined in terms of elementary topological notions. The most general setting that we want actually to use in this course is a metric space, but the propositions are so easy to state for general topological spaces that there is no reason not to do that. People who are unacquainted with topological spaces can simply think in metric spaces: reading “(X, d)” wherever one sees “(X, T)” will make things come out right. In the metric space context, to say that (X, d) and (Y, ρ) are homeomorphic under a 1-1 correspondence f , or that f is a homeomorphism of X onto Y , simply means that f : X → Y and f−1 : Y → X are continuous—but not necessarily uniformly continuous—between their respective metrics. (If f is uniformly continuous in both directions we shall continue to use the phrase uniform isomorphism.)