Categorical Interpretation
One may elegantly define the closure operator in terms of universal arrows, as follows.
The powerset of a set X may be realized as a partial order category P in which the objects are subsets and the morphisms are inclusions whenever A is a subset of B. Furthermore, a topology T on X is a subcategory of P with inclusion functor . The set of closed subsets containing a fixed subset can be identified with the comma category . This category — also a partial order — then has initial object Cl(A). Thus there is a universal arrow from A to I, given by the inclusion .
Similarly, since every closed set containing X \ A corresponds with an open set contained in A we can interpret the category as the set of open subsets contained in A, with terminal object, the interior of A.
All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example algebraic), since all are examples of universal arrows.
Read more about this topic: Closure (topology)
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