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?



Name for topological spaces where "every point has a local base wellordered by reverse inclusion"?
Note that replacing "well-ordered" by "linearly-ordered" produces an equivalent property since any linear order contains a cofinal well order.
Such spaces were called *lob-spaces* and studied by S.W. Davis in [*Spaces with linearly ordered local bases*](http://topology.nipissingu.ca/tp/reprints/v03/tp03102.pdf), Topology proceedings 3, (1978), pp.37-51.


