/sci/ - Science & Math

>>11533453
Assume you have a bunch of maps [math]f_{ \alpha } : X_{ \alpha } \rightarrow Y[/math]. They induce a set theoretical map [math]f: X \rightarrow Y[/math] the classical way.
If we have a closed set [math]A \subset Y[/math], then, for any [math]\alpha[/math], [math]f^{-1} (A) \cap X_{ \alpha }[/math] is a closed set. But [math]f^{-1} (A) = \cup _{\alpha \in I} (f^{-1}(A) \cap X_{ \alpha})[/math], which is a finite union of closed sets, and thus closed. Consequently, the preimage of closed sets is closed, and [math]f[/math] is a continuous map.
>what about the canonical part
No idea.

Maybe instead of the coproduct it's the disjoint union with the disjoint union topology? Then you could have a canonical map, I think.