Uncountable Discrete Space is not Lindelöf

Theorem

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


Then $T$ is not a Lindelöf space.


Proof

Consider the set $\CC$ of all singleton subsets of $S$:

$\CC := \set {\set x: x \in S}$

From Discrete Space has Open Locally Finite Cover, $\CC$ is an open cover of $S$ which is finer than any other open cover of $S$.

That is, $\CC$ is an open cover of $S$ which is uncountable and has no countable subcover.

(Note that a subcover is a refinement of a cover.)

So by definition $T$ can not be 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: $3$. Uncountable Discrete Topology: $8$
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