Definition Via Closure Operators
Objects: all pairs (X,cl) of set X together with a closure operator cl : P(X) → P(X) satisfying the Kuratowski closure axioms:
- (Extensivity)
- (Idempotence)
- (Preservation of binary unions)
- (Preservation of nullary unions)
Morphisms: all closure-preserving functions, i.e., all functions f between two closure spaces
- such that for all subsets of
Comments: The Kuratowski closure axioms abstract the properties of the closure operator on a topological space, which assigns to each subset its topological closure. This topological closure operator has been generalized in category theory; see Categorical Closure Operators by G. Castellini in "Categorical Perspectives", referenced below.
Read more about this topic: Characterizations Of The Category Of Topological Spaces
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