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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 !

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J.D. Wine, Paracompactifications using filter bases, Pacific J. Math. Volume 49, Number 1 (1973), 285-304, gives a number of characterizations of paracompactness in terms of $z$-filters and various kinds of filter bases. You’ll have to make your own judgement of their usefulness!

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