$X, Y$ are topological spaces and $X \times Y$ is given the product topology. The subspace $ G \subseteq X \times Y$ is defined as: $ G =$ { $(x,y) \in X \times Y | y = f(x)$ }. How can one show that if the map $f: X \to Y$ is continuous, then $G$ is homeomorphic to $X$?
Also; I have a hard time visualising what is happening. If anyone has any advise on that or knows of any good videos it would be greatly appreciated!
$G$ is just the graph of $f$, and if $f: X \to Y$ is continuous, the map $F: X \to X \times Y; F(x)=(x, f(x))$ is too, as $\pi_X \circ F = 1_X$ and $\pi_Y \circ F = f$ are both continuous by assumption and $G = F[X]$, so $F$ is a bijective continuous map from $X$ onto $G$, and it has a continuous inverse $\pi_X\restriction_G$, so $F$ is a homeomorphism.