# Product of Countably Compact Spaces is not always Countably Compact

## Theorem

Let $I$ be an indexing set.

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

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

Let each of $\struct {S_\alpha, \tau_\alpha}$ be countably compact.

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

## Proof

Let $T$ denote the Novak space.

Let $T \times T$ denote the Cartesian product of the Novak space with itself under the product topology.

From Novak Space is Countably Compact, $T$ is a countably compact space.

But from Cartesian Product of Novak Spaces is not Countably Compact, $T \times T$ is not a countably compact space.

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