Composition of functions in Munkres' Topology In Munkres' book "Topology", he writes that:

Given functions $f:A\to B$ and $g:B \to C$, .... the composition $g \circ f  $ is defined only when the range of $f$ equals the domain of $g$.

But, isn't it enough that the range of $f$ is a subset of the domain of $g$?
Why does he say that the range of $f$ needs to be equal to the domain of $g$?

Here's one good reason why Munkres requires that when composing $g \circ f$, the range of $f$ must be equal to domain of $g$ and not just a subset.  If we allowed it otherwise, it would break the theorem that says the composition of bijections is a bijection.
