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

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## Contents

## Theorem

Let $n \in \Z_{\ge 0}$ be a (strictly) positive integer.

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

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

Let each of $\struct {S_i, \tau_i}$ be weakly locally compact.

Then $\struct {S, \tau}$ is also weakly locally compact.

## Proof

## 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