Composite of Homeomorphisms is Homeomorphism

Theorem
Let $T_1, T_2, T_3$ be topological spaces.

Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be homeomorphisms.

Then $g \circ f: T_1 \to T_3$ is also a homeomorphism.

Proof
By definition of homeomorphism, $f$ and $g$ are both bijections.

From Composite of Bijections it follows that $g \circ f$ is also a bijection.

Similarly, also by definition of homeomorphism, $f$ and $g$ are both continuous mappings.

From Continuity of Composite Mapping it follows that $g \circ f$ is also a continuous mapping.

Hence the result, from definition of homeomorphism.