Countable Discrete Space is Lindelöf

Theorem

Let $T = \struct {S, \tau}$ be a countable discrete topological space.


Then $T$ is a Lindelöf space.


Proof

We have:

Countable Discrete Space is $\sigma$-Compact
$\sigma$-Compact Space is Lindelöf

So if $S$ is countable, $T$ is a Lindelöf space.

$\blacksquare$


Also see


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $2$. Countable Discrete Topology: $8$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.