Discussion
Let X be a separable first countable space, then X × X in the product topology is :
Answer: A
X × X is first countable and separable.
First countable: if \(\{U_{n}\}\) and \(\{V_{m}\}\) are countable local bases at \(x\) and \(y\), then \(\{U_{n} \times V_{m}\}\) is a countable local base at \((x, y)\).
Separable: if D is a countable dense subset of X, then D × D is countable and meets every basic open set \(U \times V\), since D meets both U and V; so it is dense in X × X.
Both properties are preserved by finite (indeed countable) products, so the options saying either may fail are wrong.
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