Exercise 2.4.7 (Subspace of a second-countable space is second-countable)

Question

Prove that a subspace of a second-countable space is second-countable.

Elu Thingol · Accepted Answer

\begin{proof}
Let $X$ be a topology with a countable base $\{U_n\}$, and let $S$ be a subset of $X$. By definition, $U_n \cap S$ is open in the relative topology of $S$ for all $n \in \mathbb{N}$. Any open set $U$ in the relative topology of $S$ can be represented as $V \cap S$ for open $V$ in $X$; thus, we can represent $U$ as a union of sets in $\{U_n \cap S\}$:
$$U = V \cap S = \left(\bigcup_i U_{n_i}\right) \cap S = \bigcup_i \left(U_{n_i} \cap S \right).$$
\end{proof}

Exercise 2.4.7 (Subspace of a second-countable space is second-countable)

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Exercise 2.4.7 (Subspace of a second-countable space is second-countable)

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