Exercise 30.8

Question

Which of our four countability axioms does $\mathbb{R}^\omega$ in the uniform topology satisfy?

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
  $\mathbb{R}^\omega$ is first countable since it is metrizable (see p.~130 and
  Example 30.2), but is not second countable by Example 30.2. By Exercise
  \href{exercise_30-5}{Exercise 30.5}, we then see that $\mathbb{R}^\omega$ does not have a countable dense subset, and is also not Lindelöf.
\end{proof}

Exercise 30.8

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Exercise 30.8

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