Theorem 2

1) If $f : X \to X'$ and $g : X' \to X''$ are two continuous mappings, then $g \circ f : X \to X''$ is continuous.

2) For a bijection $f$ of a topological space $X$ onto a topological space $X'$ to be a homeomorphism, it is necessary and sufficient that $f$ and the inverse of $f$ are continuous (or, as is also said, that $f$ is bicontinuous).

The first assertion is an immediate consequence of Proposition 2; the second follows from Theorem 1, d) and the definition of a homeomorphism (§ 1, no. 1).