Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
< Continuous Bijection from Compact to Hausdorff is Homeomorphism(Redirected from Continuous Injection from Compact Space to Hausdorff Space is Embedding)
Jump to navigation
Jump to search
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 $5.9.2$