Countable Complement Space is Lindelöf
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Theorem
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.
Then $T$ is a Lindelöf space.
Proof
By definition, $T$ is a Lindelöf space if and only if every open cover of $S$ has a countable subcover.
Let $\CC$ be an open cover of $T$.
Let $U \in \CC$ be any set in $C$.
It covers all but a countable number of points of $T$.
So for each of those points we pick an element of $\CC$ which covers each of those points.
Hence we have a countable subcover of $T$.
So by definition $T$ is a Lindelöf space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $2$