# Name for topological spaces where "every point has a local base wellordered by reverse inclusion"?

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.

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

• My friend Robert Leek has done some work that includes looking at spaces with this property. I don't know how standard his terminology is, but he refers to them as "well-based spaces" (see Definition 2.1 in arxiv.org/pdf/1401.6519.pdf). Jan 31, 2019 at 18:33
• Horst Herrlich has shown in Quotienten geordneter Räume und Folgenkonvergenz that pseudoradial spaces are exactly quotients of the spaces you describe. Jan 31, 2019 at 19:03
• If by well-ordered you really mean en.wikipedia.org/wiki/Well-order, then this implies that every point has a minimal neighborhood. Such spaces are called "Alexandroff spaces" (not to be confused with Alexandrov spaces, i.e. of metric spaces with curvature bounds). See emis.de/journals/AMUC/_vol-68/_no_1/_arenas/arenas.pdf: Arenas, F.G.. "Alexandroff spaces.." Acta Mathematica Universitatis Comenianae. New Series 68.1 (1999): 17-25 and he refers to Alexandroff P.,Diskrete Räume, Mat.Sb.(N.S.)2(1937),501–518, for the first study of such spaces. Jan 31, 2019 at 20:06
• @ClemensSämann That would be if the neighbourhoods were well-ordered by inclusion. Cla asks for neighbourhood bases well-ordered by reverse inclusion. For instance, any metric space is an example, because we can use balls of radius $\frac{1}{n}$ for $n$ a positive natural number. Please don't delete your comment, as others may have the same confusion. Jan 31, 2019 at 20:27
• Ah, sorry! I thought I checked what is meant but I mixed it up after all. Sure I let it stand as it is. Jan 31, 2019 at 20:38