Product of Countably Compact Spaces is not always Countably Compact

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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.

$\blacksquare$


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