A property between "separable" and "second countable" 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?
 A: 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 $\mathcal B$.
A: 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 $\mathcal{P}$ such that any non-empty open
subset of $X$ contains a member of $\mathcal{P}$. (The collection is downward-dense in the poset $(\mathcal{T}\setminus\{\emptyset\}, \subseteq)$ 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 $\pi\chi(x,X)$ for the minimal size of such a collection, etc.   
