Double Pointed Discrete Real Number Space is not Lindelöf
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Theorem
Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers.
Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$.
Let $T = T_\R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$.
Then $T$ is not a Lindelöf space.
Proof
We have that $T$ is a partition topology, whose basis $\PP$ is defined as:
- $\PP = \set {\set {\paren {s, a}, \paren {s, b} }: s \in \R}$
We have that $\PP$ is an open cover of $T$.
But $\PP$ has no countable subcover.
Hence the result, 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: $5$. Partition Topology: $7$