# Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?

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.

• 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/… Mar 26 at 13:38
• 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/…
– Cla
Mar 26 at 14:04
• 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.
– Cla
Mar 26 at 14:12
• 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. Mar 27 at 11:59
• I think $T_3$ in your post can be generalized to regular (where both bases are equivalent to pseudometrizable). Mar 27 at 15:07