Variation on neighbourhood base Suppose $\{\mathscr B(x) \mid x \in X\}$ is a collection of filters (or filter bases) on a set X, with each $x \in \cap\mathscr B(x)$. Then $$\mathscr T = \{U \subseteq X \mid (\forall x \in U)(\exists B \in \mathscr B(x)) B \subseteq U \} $$ defines a topology $\mathscr T$ on X. Note that $\mathscr B(x)$ is not necessarily a neighbourhood base at x, as its members need not be neighbourhoods of x.
Is there an accepted name for such collections, or any interesting known results?
(If we call a space quasi-first-countable if it has such a collection with each $\mathscr B(x)$ having a countable base then.$$\text{First-countable} \implies \text{Quasi-first-countable} \implies \text{Sequential}$$ with none of the implications being reversible. Most of that has been shown in answers to recent questions about d-metrisable spaces. (E.g d-metrizable spaces are sequential. ) Quasi-first-countable is equivalent to d-metrisable.)
 A: tl;dr; I would call such collection a (local) network system.
Let $X$ be a topological space, $x ∈ X$. We call a nonempty collection $\mathscr{B}(x) ⊆ \mathscr{P}(X)$ such that $(∀B ∈ \mathscr{B}(x))\ x ∈ B$.


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* (local) network at $x$ if for every neighborhood $U$ of $x$ there is some $B ∈ \mathscr{B}(x)$ such that $x ∈ B ⊆ U$;

* (local) (neighborhood} base at $x$ if in addition every $B ∈  \mathscr{B}(x)$ is a neighborhood of $x$;

* (local) open (neighborhood) base at $x$ if in addition every $B ∈  \mathscr{B}(x)$ is open.


Then the corresponding mapping $\mathscr{B} := \{x \mapsto \mathscr{B}(x): x ∈ X\}$ is called (local) network system, (local) neighborhood system, or (local) open neighborhood system, respectivelly. One can observe that the topology is uniquely determined by such system. As you write, $\mathscr T_\mathscr{B} = \{U \subseteq X \mid (\forall x \in U)(\exists B \in \mathscr B(x))\ B \subseteq U \}$.
On the other hand we may start with a set $X$ and a such mapping $\mathscr{B}$ and ask, when $\mathscr{T}_\mathscr{B}$ is a topology and under what conditions $\mathscr{B}$ is its network system or even (open) neighborhood system.
When we have a set $X$ and a mapping $\mathscr{B}$, by the topology generated by the network / neighborhood / open neighborhood system, we mean just $\mathscr{T}_\mathscr{B}$ and the claim that $\mathscr{B}$ is a corresponding system of induced topology.
So I would say, that $X$ is first-countable if it is generated by some neighborhood system $\mathscr{B}$ such that $\mathscr{B}(x)$ is countable for each $x$. And similarily $X$ is quasi-first-countable if it is generated by some network system $\mathscr{B}$ such that $\mathscr{B}(x)$ is countable for each $x$.
