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One approach to define a topology (instead of open/closed sets) is by the Kuratowski Closure Operator. This operator $cl$ uniquely defines a topology (defined by open/closed sets). But since every topological space has a basis, trivially, I wonder how one may define a basis in terms of the Kuratowski Closure Operator (or its dual, the Interior Operator). Given the fact that any topology can be defined by such an operator, can a basis for a topology be described by e.g. relaxing some of the closure operator axioms?

Background: A topology can be constructed equally by a collection of filters (see my question and the answers here: Characterization of Topology). Since there exists a notion of a filter basis one can construct — based on the equivalence proved in the link — a definition of a basis for a topology in terms of a filter. The resulting basis is known as local basis.

My question is basically wether there a) exists a similar notion for closure/interor operators and if not, b) is it possible to define one?

Syd Amerikaner
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  • The closure operator creates a topology which is the largest base. What does "bases should be either" mean? – William Elliot Oct 16 '18 at 23:58
  • thanks for your reply. I added a little more context. – Syd Amerikaner Oct 17 '18 at 00:21
  • Is there such a thing as a base for a closure operator like there is for filters? – William Elliot Oct 17 '18 at 08:55
  • For base we already have a fine characterisation using the two well-known axioms. We're not constructing a topology from a filter, but from a system of filters (quite a different beast), one for each point. If you want to have filter bases in instead of filters in that approach, some axioms need to be changed and some proofs too. Left to the interested reader. – Henno Brandsma Oct 17 '18 at 23:19
  • Let me see if I understand you: do you want an interior operator axioms such that the fix points of that operator are precisely a base for some topology? Please clarify, or the question is (IMHO) too vague. – Henno Brandsma Oct 17 '18 at 23:21
  • @HennoBrandsma Since there exists a notion of a filter basis, defining a topology by filters immediately gives raise to defining a basis for a topology by filter a filter basis.

    And since a Kuratowski closure operator also defines a topology, I wonder wether there is something like a "Kuratowski closure operator basis", i.e. a set of axioms that somehow help defining such a closure operator (like what a filter basis is to a filter). I hope it is more clear now.

     – Syd Amerikaner Oct 21 '18 at 21:20

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