First-Countability is Hereditary
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Theorem
Let $T = \struct {S, \tau}$ be a topological space which is first-countable.
Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.
Then $T_H$ is first-countable.
Proof
From the definition of first-countable, every point in $S$ has a countable local basis in $T$.
The intersection of $H$ with the countable local basis of $S$ provides a countable local basis for $H$.
As every point in $H$ is also a point in $S$, 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