Countable Product of Second-Countable Spaces is Second-Countable
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Theorem
Let $\left \{{\left({X_\alpha, \vartheta_\alpha}\right)}\right\}$ be a countable set of topological spaces.
Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \{{\left({X_\alpha, \vartheta_\alpha}\right)}\right\}$.
Let each of $\left({X_\alpha, \vartheta_\alpha}\right)$ be second-countable.
Then $\left({X, \vartheta}\right)$ is also second-countable.
Proof
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 3$: Invariance Properties
