Subspace of Either-Or Space less Zero is not Lindelöf
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be the either-or space.
Let $H = S \setminus \set 0$ be the set $S$ without zero.
Then the topological subspace $T_H = \struct {H, \tau_H}$ is not a Lindelöf space.
Proof
By definition of topological subspace, $U \subseteq H$ is open in $T_H$ if and only if:
- $(1): \quad \set 0 \nsubseteq U$
or:
- $(2): \quad \openint {-1} 1 \subseteq U$
But for all $U \subseteq H$, condition $(1)$ holds as $0 \notin H$.
So $T_H$ is by definition a discrete space.
As $T_H$ is uncountable, we have that Uncountable Discrete Space is not Lindelöf holds.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $17$. Either-Or Topology: $2$