Second-Countability is Hereditary
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Theorem
Let $T = \struct {S, \tau}$ be a topological space which is second-countable.
Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.
Then $T_H$ is second-countable.
Proof
From the definition of second-countable, $\struct {S, \tau}$ has a countable basis.
That is, $\exists \BB \subseteq \tau$ such that:
As $H \subseteq S$ it follows that a $H$ itself is a union of sets from $\BB$.
The result follows from Basis for Topological Subspace.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Countability Properties