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Call a topological space $\langle X,\mathscr{O}\rangle$ regular iff it is both $T_0$ and $T_3$: for every point $x\notin A$, where $A$ is a closed subsets of $X$, there are open and disjoint sets $V$ and $U$ such that $x\in V$ and $A\subseteq U$.

A $\langle X,\mathscr{O}\rangle$ space is linear-based iff in every its point there is a local basis which is linearly ordered by $\subseteq$ relation.

Could you give me an example of a regular space which is not linear-based?

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  • $\begingroup$ Please provide the information about where in mathematics do arrive mentioned above by you linear-based topological spaces, why are them interesting? $\endgroup$
    – Evgeny Kuznetsov
    Commented Apr 26, 2018 at 9:06
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    $\begingroup$ Clearly all metric spaces are linear based. Thus not being linear based is an obstruction to metrizability. The example I gave is an example that the second countability assumption in Urysohns metrization theorem is essential. $\endgroup$
    – HenrikRüping
    Commented Apr 26, 2018 at 10:10
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    $\begingroup$ I am not aware of it. $\endgroup$
    – HenrikRüping
    Commented Apr 26, 2018 at 10:27
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    $\begingroup$ In my case the interest in such spaces stems from point-free topology, especially the part of it dealing with the so-called Boolean Contact Algebras. Point-free counterparts of linear bases (i.e. certain chains in BCAs) are good approximations of points and linear-based spaces are used for representation theorems for BCAs (at least in my studies of BCAs, there are different approaches too). As far as I know there is no book nor paper devoted to class of linear-based spaces. But check the so-called well-based spaces and radial spaces which are related to linear-based. $\endgroup$
    – Rafał Gruszczyński
    Commented Apr 26, 2018 at 10:44
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    $\begingroup$ @EvgenyKuznetsov This freely available paper by Dimiter Vakarelov may be a good starting point: link.springer.com/chapter/10.1007%2F978-0-387-69245-6_6 $\endgroup$
    – Rafał Gruszczyński
    Commented Apr 26, 2018 at 14:11

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An uncountable product of copies of the unit interval or of the discrete two element space. Indeed none of the points in this space has a linearly ordered local basis.

Let us have a look at the latter. Let $X:= \prod_I \{0,1\}$, let $0\in X$ be the sequence which is zero everywhere and let $S_i\subset X$ be the subset consisting of all sequences whose $i$-th coordinate is zero.

By definition of the product topology, any open neighborhood of $0$ can be contained in only finitely many of the $S_i$'s.

Now suppose you have a linearly ordered basis $B$. Assign to each basis element $U$ the finite subset $f(U) = \{i\in I| S_i\supset U\}$.

Since $B$ is a basis, we know that every $i$ appears in some $f(U)$. It follows that $I$ is an union of nested, finite sets and hence countable. Contradiction!

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  • $\begingroup$ Henrik, the third paragraph should say "any basic open neighborhood of 0", for if you take for example $T_n$ to be the set of all sequences which have 0's at first $n$ terms, than $\bigcup_{n\in\omega}T_n$ is cotained in every $S_i$, for $i\in\omega$. Am I right? $\endgroup$
    – Rafał Gruszczyński
    Commented Nov 20, 2018 at 22:03
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    $\begingroup$ @MadHatter: The $T_i$'s form a decreasing sequence of open sets, Thus their union is just the first of the $T_i$'s, if we replaced union by intersection, we dont get something open. It also follows that if any basic open neighborhood of 0 is contained in only finitely many of the $S_i$'s, then any open nbhd of 0 can be contained in finitely many of the $S_i$'s (just pick a basic open nbh contained in it). $\endgroup$
    – HenrikRüping
    Commented Nov 21, 2018 at 3:20

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