Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary

Corollary to Continuous Bijection from Compact to Hausdorff is Homeomorphism

Let $T_1$ be a compact space.

Let $T_2$ be a Hausdorff space.

Let $f: T_1 \to T_2$ be a continuous injection.

Then $f$ determines a homeomorphism from $T_1$ to $f \sqbrk {T_1}$.

That is, $f$ is an embedding of $T_1$ into $T_2$.

Proof

Follows from Continuous Bijection from Compact to Hausdorff is Homeomorphism and Continuity of Composite with Inclusion.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.9$: An inverse function theorem: Theorem $5.9.2$