7
$\begingroup$

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?

$\endgroup$
6
  • 1
    $\begingroup$ 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
    Jan 31, 2019 at 18:33
  • 3
    $\begingroup$ Horst Herrlich has shown in Quotienten geordneter Räume und Folgenkonvergenz that pseudoradial spaces are exactly quotients of the spaces you describe. $\endgroup$ 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. $\endgroup$ Jan 31, 2019 at 20:06
  • 4
    $\begingroup$ @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. $\endgroup$ 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. $\endgroup$ Jan 31, 2019 at 20:38

1 Answer 1

9
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ This paper seems really what I was looking for. Thank you, I'll take my time to read it. $\endgroup$
    – Cla
    Feb 1, 2019 at 14:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.