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In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?


A bit of context:

Given a topological space $X$, a family $\mathcal{U}\subseteq \mathcal{P}(X)$ is locally finite (resp., discrete) if for every $x\in X$ there is an open neighborhood $U$ of $x$ that intersects only finitely many (resp., at most one) elements of $\mathcal{U}$.

We say $\mathcal{U}$ is $\sigma$-locally finite (resp., $\sigma$-discrete) if it can be partitioned into countably many locally finite (resp., discrete) families.

Nagata-Smirnov-Bing metrization theorem states that the following are equivalent:

  1. $X$ is metrizable,
  2. $X$ is $T_3$ and has a $\sigma$-locally finite basis,
  3. $X$ is $T_3$ and has a $\sigma$-discrete basis.

In particular, the two properties (having a $\sigma$-locally finite basis, and having a $\sigma$-discrete basis) are equivalent for $T_3$ spaces.

On the other hand, there are examples of ($T_0$) spaces having a $\sigma$-locally finite basis but no $\sigma$-discrete basis, e.g. the poset $([\omega_1]^{<\omega}, \supseteq)$ with the Alexandrov topology.

So the two properties are not equivalent in general.

I tried searching but I could not find any Hausdorff counterexample.

My question is: are the two properties still equivalent for $T_2$ spaces?


Note:

Say that $\mathcal{U}\subseteq \mathcal{P}(X)$ is star-finite if every $U\in\mathcal{U}$ intersects only finitely many elements of $\mathcal{U}$.

We say $\mathcal{U}$ is $\sigma$-star-finite if it can be partitioned into countably many star-finite families.

Since every star-finite family is also $\sigma$-discrete, we get the following

Remark: $\sigma$-discrete $\iff$ $\sigma$-star-finite $\Rightarrow$ $\sigma$-locally finite.

In particular, if a space has a $\sigma$-locally finite basis but not a $\sigma$-discrete basis, the locally finite families should be very far from star-finite.

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    $\begingroup$ Lastly, here are all the spaces in $\pi$-base that are Hausdorff, have a $\sigma$-locally finite basis, and are not $T_3$. But, I can't seem to filter the search on $\sigma$-discrete basis. If you know some equivalent characterizations of that notion, maybe the answer is in $\pi$-base. topology.pi-base.org/… $\endgroup$ Mar 26 at 13:38
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    $\begingroup$ Thanks a lot, I did not know that $\pi$-base had also a filter for spaces with a $\sigma$-locally finite basis! Unfortunately though, it seems that it does not contain any example of a space that has a $\sigma$-locally finite basis and is $T_2$ bu not $T_3$, nor second countable (you want to avoid second countable spaces as they are always $\sigma$-discrete). topology.pi-base.org/… $\endgroup$
    – Cla
    Mar 26 at 14:04
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    $\begingroup$ The first math.stackexchange link does not seem very related. The second one instead seems related (that is basically the idea for the $T_0$ counterexample I wrote in the question) but I could not exploit that idea to obtain a Hausdorf space. $\endgroup$
    – Cla
    Mar 26 at 14:12
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    $\begingroup$ Every example in the pi-Base of a space with $\sigma$-locally finite base was deduced automatically from it being second countable or pseudometrizable topology.pi-base.org/properties/P000054/spaces it'd be great to add an example or theorem to satisfy Cla's search. $\endgroup$ Mar 27 at 11:59
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    $\begingroup$ I think $T_3$ in your post can be generalized to regular (where both bases are equivalent to pseudometrizable). $\endgroup$ Mar 27 at 15:07

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