Uncountable Discrete Space is not Lindelöf
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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: $8$