Let $T_X = (X, \mathcal{T}_X)$ and $T_Y = (Y, \mathcal{T}_Y)$ each be a D1106: Topological space.
A D18: Map $f : X \to Y$ is a homeomorphism from $T_X$ to $T_Y$ if and only if

(1)	$f$ is a D55: Continuous map from $T_X$ to $T_Y$
(2)	$f$ is an D976: Invertible map with an D216: Inverse map $f^{-1} : Y \to X$
(3)	$f^{-1}$ is a D55: Continuous map from $T_Y$ to $T_X$