Let $(X, \tau)$ be a topological space. It is second countable if it has a countable basis $B \subseteq \tau$. It is separable if there exists a countable $S \subseteq X$ such that $O \cap S \neq \emptyset$ for every nonempty $O \in \tau$. It is well known that second countability is strictly stronger than separability.
I'm working on something hinges on an intermediate property: "there exists a countable subset $C \subseteq \tau$ [edit: with each $C$-member nonempty!] that is dense in $\tau$, in the sense that for all $O \in \tau$, there exists $P \in C$ such that $P \subseteq O$."
Is there a common name for this property? I will call it "property C" for now.
Second countability implies property C (since a countable basis for $\tau$ is dense in $\tau$), which implies separability (choose one member from each $P \in C$ and the set of all the choices serves as the $S$ in the definition of separability). The Moore plane is an example of a topology that has property C but is not second countable.
Are there examples of topological spaces that are separable but do not have property C?
Best Answer
Consider $\mathbb R$ with the finite complement topology. Any infinitely countable subset $A$ of $\mathbb R$ is dense since an open subset of $\mathbb R$ can only miss finitely many points of $A$.
Let $\mathcal B$ be a countable family of non-empty open subsets of $\mathbb R$. Then $\bigcap_{B \in \mathcal B} B$ misses countably many points of $\mathbb R$ at most. It follows that some $x \in \mathbb R$ lies in every element of $\mathcal B$. Thus, the open subset $\mathbb R - \{x\}$ does not contain any element of $\mathcal B$.