# Weak Local Compactness is Preserved under Open Continuous Surjection

## Contents

## Theorem

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ 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 weakly locally compact, then $T_B$ is also weakly locally compact.

## Proof

Let $\phi$ be a mapping which is surjective, continuous and open.

Let $T_A$ be weakly locally compact.

Take $b \in S_B$.

Let $V$ be a neighbourhood of $b$.

Since $\phi$ is surjective:

- $\forall y \in S_B: \exists x \in S_A: x \in \map {\phi^{-1} } y$

From the weak local compactness of $T_A$ and the continuity of $\phi$, there exists a compact neighbourhood $K$ of $x$ such that $\phi \sqbrk K \subseteq V$.

Since $K$ is a neighbourhood of $x$, then $x \in K^\circ$ and $y \in \phi \sqbrk {K^\circ} \subseteq \phi \sqbrk K$, where $K^\circ$ is the interior of $K$.

$\phi$ is an open mapping and $K^\circ$ is an open set, so $\phi \sqbrk {K^\circ}$ is also open.

Finally we get that $y \in \phi \sqbrk K \subseteq V$, where $\phi \sqbrk K$ is a compact neighbourhood.

Thus $T_B$ is weakly locally compact.

$\blacksquare$

## Also see

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Invariance Properties