Continuity of functions between metric spaces question.

1. Dec 30, 2011

DavidAlan

Was trying to learn differential geometry, had to take time off of it to develop some knowledge of topology, namely compactness and Hausdorff's condition. I'm using Sutherland's book on topology and came across something I didn't understand concerning metric spaces,

Sutherland speaks of the so called (d,d')-continuity of a function restricted to a subset of a metric space.

If you have two metric spaces, {A, d} and {A', d'} as well as a map f: A→A' and a metric d$_{H}$ induced on a subspace H of A by A, then apparently there is some notion of the (d$_{H}$, d')-continuity of f|H at h$\in$H that must be distinguished from normal continuity of f at h.

Perhaps someone can explain this in more depth, Sutherland introduces this in passing and then goes on to use it to define topological equivalence (as an equivalence relation). I have only a very naive understanding of what a homeomorphism is, maybe this is the key for me to understand it rigorously.