The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies point the same way as in the diagrams for the product and disjoint union topologies, respectively (but there are different conditions on the "constructor" arrow). This leads me to wonder:



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Categorical Construction of Quotient Topology?
Inclusions of subspaces are precisely the regular monomorphisms, and projections of quotients are precisely the regular epimorphisms.


