Does non-standard analysis still make use of classical topology?

I understand that non-standard analysis is generally viewed as an alternative to the $\epsilon$-$\delta$ approach of classical analysis.

So for example, I'm guessing that the definition of a continuous function between metric spaces would be made rigorous using infinitesimals, rather than the usual $\epsilon$-$\delta$ definition.

However, the notion of continuity can be generalized to functions between arbitrary topological spaces.

My question is, how does non-standard analysis stand in relation to general topology? Are concepts like "continuous function between topological spaces" still defined and studied, and if so, are they defined in the same way as in the classical approach?

What about compactness and connectedness? Are they defined in the same way, or are they defined differently?

More generally, how much of a person's knowledge about classical analysis can be expected to transfer over to the non-standard approach?

• conitnuity is defined but not equivalent to the classical definition (functions that are classically not continuous might be nonstandard-continuous). Compactness and connectedness are defined and very similar if not identical. My knowedge of nonstandard analysis is very basicc though and i will let someone more knowledgable than me answer this. – example Feb 15 '13 at 23:52

• If non-standard analysis merely "enriches" classical analysis, how does it do away with $\epsilon-\delta$ arguments? – goblin Feb 17 '13 at 2:36
• It allows to replace $\epsilon - \delta$ arguments with other arguments. – Ittay Weiss Feb 17 '13 at 2:49
• Nonstandard analysis does not "do away with" $\epsilon$-$\delta$ argument any more than arguments using compactness or passage to subsequences do. Like these other things, it simply provides additional tools, including further characterizations of key concepts like limits and continuity. It is not a "rival" to standard analysis in any mathematical sense. – Pete L. Clark Aug 4 '13 at 16:16