Uncountable Particular Point Space is not Lindelöf
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Theorem
Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.
Then $T$ is not a Lindelöf space.
Proof
Consider the open cover of $T$:
- $\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is uncountable, then so is $\CC$, as we can set up a bijection $\phi: S \setminus \set p \leftrightarrow \CC$:
- $\forall x \in S \setminus \set p: \map \phi x = \set {x, p}$
Hence $\CC$ has no countable subcover.
The result follows by definition of Lindelöf space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $10$. Uncountable Particular Point Topology: $5$