Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact

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.