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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 $O in tau$. 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?

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 $mathcal B$.

What you're looking for is the concept of a $pi$-base (or pseudobase), i.e. a collection of non-empty (this matters!) open subsets $mathcalP$ such that any non-empty open
subset of $X$ contains a member of $mathcalP$. (The collection is downward-dense in the poset $(mathcalTsetminusemptyset, subseteq)$ is another way of putting it)

The minimal size of a $pi$-base for $X$ is denoted $pi w(X)$ (rounded up to $aleph_0$ if necessary, in Juhasz it's $pi(X)$) , see the cardinal functions sections of this wikipedia page. So property $C$ is countable $pi$-weight or $pi w(X)=aleph_0$ in more conventional terms, and I believe property C is already taken as a name in topology, or at least property (K) is, for sure. (which has the related meaning that every uncountable set of open subsets has an uncountable subset that pairwise intersect; a property implied by but weaker than separability). I prefer countable $pi$-weight, or having a countable $pi$-base as a name, being a bit more descriptive.

As to examples: For an $X$ just $T_1$ but not higher, the cofinite topology on an uncountable $X$ is separable and does not have a countable $pi$-base. A more advanced example (compact Hausdorff): $[0,1]^Bbb R$ in the product topology is separable but has no countable $pi$-base, as a counting argument involving basic subsets will reveal. That both examples are not first countable is no accident: if $X$ is both separable and first countable, the union of the local bases at the countable dense subset form a countable $pi$-base, as is easily checked. For metric spaces, having a countable base, being separable and having a countable $pi$-base are all equivalent.

There also exists the notion of a local $pi$-base at $x$: a collection of non-empty open subsets of $X$ such that every neighbourhood of $x$ contains a set from it. This is related to notions like tightness at a point etc. We get a similar cardinal invariant of $pichi(x,X)$ for the minimal size of such a collection, etc.