Finite Product of Locally Compact Spaces is Locally Compact

Theorem

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

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

Let $\displaystyle \left({X, \vartheta}\right) = \prod_{i=1}^n \left({X_i, \vartheta_i}\right)$ be the product space of $\left \{{\left({X_i, \vartheta_i}\right): 1 \le i \le n}\right\}$.

Let each of $\left({X_i, \vartheta_i}\right)$ be locally compact.

Then $\left({X, \vartheta}\right)$ is also locally compact.

Sources

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