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?