# Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact

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## Contents

## Theorem

Let $n \in \Z_+^*$ be a positive integer.

Let $\left\{ {\left({S_i, \tau_i}\right): 1 \le i \le n}\right\}$ be a finite set of topological spaces.

Let $\displaystyle \left({S, \tau}\right) = \prod_{i \mathop = 1}^n \left({S_i, \tau_i}\right)$ be the product space of $\left\{ {\left({S_i, \tau_i}\right): 1 \le i \le n}\right\}$.

Let each of $\left({S_i, \tau_i}\right)$ be weakly locally compact.

Then $\left({S, \tau}\right)$ is also weakly locally compact.

## Proof

## Also see

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 3$: Invariance Properties