There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base.

Is there a similar name for a space where "every point has a local base wellordered by reverse inclusion"?

Does this property have any known consequence or relation with other properties? Clearly first countable implies this properties, but I would be more interested in not-first-countable spaces.

There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base.

Is there a similar name for a space where "every point has a local base wellordered by reverse inclusion"?

That is, given $X=(X,\tau)$ topological space, we say $X$ has the property if for every $x\in X$ there exists $\mathcal{B}=\{B_i\}_{i\in\kappa}\subset\tau$ such that $B_i\subset B_j$ for every $j<i<\kappa$.

Does this property have any known consequence or relation with other properties?