Pullback in $\mathrm{TOP}$

Question

The pushout in the category of topological spaces is given by the gluing of spaces along continuous maps. Does there exist a similar "easy" topological description of the pullback? Does it even always exist in this category?

Answer 1

Yes, exists, and is a subspace of $A\times B$, in case you want to pull back $f:A\to C$ and $g:B\to C$. Namely, a kind of 'equalizer': $$A{}_f\times_g B:= \{ (a,b)\mid f(a)=g(b)\} $$ I think you can draw it somehow..

Somehow.. generally, pullback 'asks' for the solution (when $f=g$?), and pushout 'realizes' or 'forces' (topologically: 'glues') the solution, saying: "let $f(a)=g(a)$ in the new space for all $a$".