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?

Answer

Nonstandard techniques, by means of some logical tools (usually either the compactness theorem or ultrapower construction) can and are applied to many areas including metric space theory, topology, measure theory, lattice theory, and category theory.

Nonstandard topology enriches ordinary topology in much the same way as nonstandard calculus enriches ordinary calculus. The familiar definitions from topology can be extended to nonstandard notions, the transfer principle applies, and typical arguments in topology can be replaced by nonstandard arguments.

Regarding connectedness there is little difference between ordinary and nonstandard topology. Interestingly, the notion of compactness is illuminated differently with nonstandard analysis in what is considered to be among the nicest immediate consequences of nonstandard techniques in topology. The result is due to Robinson and states that a topological space is compact if, and only if, in an enlargement every point is infinitesimally close to a standard point.

Regarding you last question, the transfer principle answers that precisely. Any first order property is true classically if, and only if, it is true in an enlargement.