Countable topological basis with some inclusion properties

Question

Given a second countable topological space $(X,\tau)$, I want a countable basis $\mathcal{B}$ with the following properties: \begin{align}&\forall B \in \mathcal{B} \text{ the set } \{B' \in \mathcal{B} \mid B \subseteq B'\} \text{ is finite} \tag{1}\\ &\forall B \in \mathcal{B} \text{ the set } \{B' \in \mathcal{B} \mid B \subseteq B'\} \text{ is well-founded wrt } \supseteq \tag{2}\end{align} my questions are:

1. Is there a name for such a properties ?
2. Can we find both kinds of basis for every second countable space?
3. Otherwise, is there a counterexample?
4. Are there other well-known topological properties of the space that guarantee the existence of such basis (or that is equivalent to the existence of such a basis)?

Thanks!

EDIT: Note that condition (2) is (strictly?) weaker than (1). Indeed condition (2) says that there are no infinite increasing wrt $\subseteq$ sequences of elements of the basis.

Answer 1

Look at the topology $\mathcal{T}$ on $[0,1)$ generated by the basis $[0,q)$.

Any countable basis $B$ of $\mathcal{T}$ will not satisfy your first criterion. Or your second criterion.