Local Compactness is Preserved under Open Continuous Surjection

Theorem

Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be topological spaces.

Let $\phi: T_A \to T_B$ be a continuous mapping which is also an open mapping and a surjection.


If $T_A$ is locally compact, then $T_B$ is also locally compact.


Proof


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Also see


Sources

  • 1970: Stephen Willard: General Topology: Chapter $6$: Compactness: $\S18$: Locally Compact Spaces: Theorem $18.5$
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