Definition talk:Continuous Mapping (Metric Space)/Space/Definition 2

It appears that this definition is using different words to say that a mapping between metric spaces is continuous iff it is continuous with respect to their induced topologies. That form is substantially more important, I believe. Is it doomed to forever sit at definition #3 once I create it? --Dfeuer (talk) 01:30, 11 January 2013 (UTC)

I personally do not see how a separate page is merited. I think it suffices to add a sentence such as:

By definition, this is equivalent to the continuity of $f$ with respect to the induced topologies on $A_1$ and $A_2$.

Comments? --abcxyz (talk) 01:58, 11 January 2013 (UTC)

That makes a lot of sense. Huzzah. --Dfeuer (talk) 02:19, 11 January 2013 (UTC)

I guess I'll just add that and see if any objections are raised or not. --abcxyz (talk) 02:22, 11 January 2013 (UTC)

A few steps left now to refactor (down to nothing) the proof that metric space completeness is invariant under isometry. Still need to show that an isometry is homeomorphism (trivial now that we have this result), and that sequence convergence in a metric space has a similarly equivalent definition in terms of the induced topology. --Dfeuer (talk) 02:32, 11 January 2013 (UTC)