Abstract
The axioms for a topology in terms of open sets follow necessarily from the intuitive relation of this concept with ultrafilter convergence. By contrast, the intuitive relations between neighbourhood systems or closure operations on the one hand and ultrafilter convergence on the other lead only to pretopologies. Kleisli compositions, previously used in categorical algebra, greatly facilitate categorical descriptions of topological spaces, both in terms of neighbourhood systems and (ultra)filter convergence relations.
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