Countable Excluded Point Space is Separable
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space.
Then $T$ is a separable space.
Proof
The closure of the set $S$ (trivially) equals $S$.
That is, $S$ is everywhere dense in $T$.
But as $S$ is countable it follows by definition that $T$ is separable.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $14$. Countable Excluded Point Topology: $6$