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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?

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    $\begingroup$ 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). $\endgroup$
    – Will Brian
    Commented Jan 31, 2019 at 18:33
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    $\begingroup$ Horst Herrlich has shown in Quotienten geordneter Räume und Folgenkonvergenz that pseudoradial spaces are exactly quotients of the spaces you describe. $\endgroup$
    – Martin Sleziak
    Commented Jan 31, 2019 at 19:03
  • $\begingroup$ 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. $\endgroup$
    – Clemens Sämann
    Commented Jan 31, 2019 at 20:06
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    $\begingroup$ @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. $\endgroup$
    – Robert Furber
    Commented Jan 31, 2019 at 20:27
  • $\begingroup$ Ah, sorry! I thought I checked what is meant but I mixed it up after all. Sure I let it stand as it is. $\endgroup$
    – Clemens Sämann
    Commented Jan 31, 2019 at 20:38

1 Answer 1

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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, Topology proceedings 3, (1978), pp.37-51.

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  • $\begingroup$ This paper seems really what I was looking for. Thank you, I'll take my time to read it. $\endgroup$
    – Cla
    Commented Feb 1, 2019 at 14:38
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    $\begingroup$ I bet I can guess what the letters "lob" stand for ... $\endgroup$
    – Nik Weaver
    Commented Apr 28, 2023 at 12:13
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    $\begingroup$ The same notion appears in N. Howes, A note on transfinite sequences, Fundamenta Mathematicae 106 (1980), 213-226, under the name "chain local base (CLB) space". $\endgroup$
    – PatrickR
    Commented Oct 24, 2023 at 4:58

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