Deleted Integer Topology is First-Countable
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Corollary to Deleted Integer Topology is Second-Countable
Let $S = \R_{\ge 0} \setminus \Z$.
Let $\tau$ be the deleted integer topology on $S$.
The topological space $T = \struct {S, \tau}$ is first-countable.
Proof
From Deleted Integer Topology is Second-Countable, $T$ is second-countable.
The result follows from Second-Countable Space is First-Countable.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $7$. Deleted Integer Topology: $5$