Definition Via Closed Sets
Objects: all pairs (X,T) of set X together with a collection T of subsets of X satisfying:
- The empty set and X are in T.
- The intersection of any collection of sets in T is also in T.
- The union of any pair of sets in T is also in T.
- The sets in T are the closed sets.
Morphisms: all functions such that the inverse image of every closed set is closed.
Comments: This is the category that results by replacing each lattice of open sets in a topological space by its order-theoretic dual of closed sets, the lattice of complements of open sets. The relation between the two definitions is given by De Morgan's laws.
Read more about this topic: Characterizations Of The Category Of Topological Spaces
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