Infinite Product of Weakly Locally Compact Spaces is not always Weakly Locally Compact

Theorem

Let $I$ be an indexing set with infinite cardinality.

Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$.

Let each of $\struct {S_\alpha, \tau_\alpha}$ be weakly locally compact-compact.

Then it is not necessarily the case that $\struct {S, \tau}$ is also weakly locally compact.

Proof

Let $T = \struct {\Z_{\ge 0}, \tau}$ be the topological space formed by the discrete topology on the set of positive integers.

Let $T' = \struct {\ds \prod_{\alpha \mathop \in \Z_{\ge 0} } \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the countable Cartesian product of $\struct {\Z_{\ge 0}, \tau}$ indexed by $\Z_{\ge 0}$ with the product topology $\tau'$.

From Discrete Space is Strongly Locally Compact, $T$ is strongly locally compact.

From Strongly Locally Compact Space is Weakly Locally Compact, $T$ is weakly locally compact.

From Countable Product of Countable Discrete Spaces is not Weakly Locally Compact, $T'$ is not weakly locally compact.

Hence the result.

$\blacksquare$

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