Double Pointed Fortissimo Space is Lindelöf
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a Fortissimo space.
Let $T \times D$ be the double pointed topology on $T$.
Then $T \times D$ is a Lindelöf space.
Proof
Let $D = \set {0, 1}$.
Let $\CC$ be an open cover of $T \times D$.
Then $\exists A \times B \in \CC$ such that $\tuple {p, 0} \in A \times B$.
We must have $\relcomp S A$ is countable and $B = D$.
Hence $\relcomp S A \times D$, a product of countable sets, must be countable.
So $A \times D$, together with an open neighborhood of each of the elements of $\relcomp S A \times D$, is a countable subcover of $\CC$.
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: $25$. Fortissimo Space: $4$