If one use ultrafilters to describe a compact space, one gets Tychonoff Theorem as a trivial result. So, im just asking if there is such a "useful" equivalence, but concerning paracompact spaces ( e.g. Smirnov-Theorem in the language of filters, since it has Hausdorff condition - any filters converges at most to one point - and the paracompactness condition - "some useful description of those spaces with filter")
Thanks !