Let $f:X\to Y$ be an application of topologies. Consider the set $$\Gamma_f=\{(x,f(x))|x\in X\} \subset X \times Y$$ with the product topology on $X \times Y$.
Prove that $f$ is continuous $\iff$ $\Psi:X \to \Gamma_f, \Psi(x)=(x,f(x))$ is an homeomorphism.
I have a question about the implication "$\Longleftarrow$":
If $\Psi$ is an homeomorphism, in particular $\Psi$ is a continuous application. Now ,assuming that $X$ and $Y$ are topologies induced by the distances $d_1$ and $d_2$, for the continuity of $\Psi$ I have $\forall \epsilon>0$ $\exists \delta>0:$ if $d_1(x,\bar x)<\delta\implies d_2(\Psi(x),\Psi(\bar x))<\epsilon$ this implies the continuity of $f$. The problem is that I am supposing that $X$ and $Y$ are metric topologies… Is there a more general way to prove this?
Best Answer
In this case, $f$ is continuous as a composition of continuous functions:
$$f:X\overset{\Psi}\to\Gamma_f\overset{\text{inc}}\hookrightarrow X\times Y\overset{\text{pr}_2}\to Y$$