A space $X$ is:
- Lindelof if every open cover for $X$ has a countable subcover.
- A $P$-space if every $G_\delta$ subset of $X$ is open.
- Discretely generated if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$, there is a discrete set $D \subset A$ such that $x \in \overline{D}$.
- Weakly discretely generated if for every non-closed set $A \subset X$ there is a discrete set $D \subset A$ such that $\overline{D} \setminus A \neq \emptyset$.
(All spaces are assumed to be Hausdorff).
A compact space is weakly discretely generated. This is an immediate consequence of the following well-known fact:
A space $X$ is compact if and only if the closure of every discrete subset of $X$ is compact.
For a proof see:
- Tkachuk, V. V., Spaces that are projective with respect to classes of mappings, Trans. Mosc. Math. Soc. 1988, 139-156 (1988); translation from Tr. Mosk. Mat. O.-va 50, 138-155 (1987). ZBL0662.54007. or
- Arhangel’skij, A. V., A generic theorem in the theory of cardinal invariants of topological spaces, Commentat. Math. Univ. Carol. 36, No. 2, 303-325 (1995). ZBL0837.54005.
There is no reason to believe that what's true for compact spaces must also hold for Lindelof spaces and indeed it's relatively easy to find Lindelof (even countable) spaces which are not weakly discretely generated. It suffices to take a countable maximal space (see my other question What's the minimal weight of a maximal space?).
However, Lindelof $P$-spaces resemble compact spaces much more than general Lindelof spaces (for example, Lindelof regular $P$-spaces are normal, a Lindelof subspace of a Lindelof $P$-space must be closed and at least countable products of Lindelof $P$-spaces are Lindelof whereas the product of two Lindelof spaces can even fail to be normal, etc...).
QUESTION 1: Is it true that every Lindelof $P$-space is weakly discretely generated?
The above question was first formulated, albeit not explicitly in Bella, Angelo; Simon, Petr, Spaces which are generated by discrete sets., Topology Appl. 135, No. 1-3, 87-99 (2004). ZBL1050.54001.
Actually, even the following question is still open:
QUESTION 2: Is it true that every Lindelof $P$-space is discretely generated?
A negative answer to QUESTION 1 would also solve this other Mathoverflow Question:
https://mathoverflow.net/questions/140224/is-there-a-linearly-lindelöf-non-lindelöf-p-space
That's because of the following proposition from Alas, Ofelia T.; Junqueira, Lucia R.; Wilson, Richard G., When is a $P$-space weakly discretely generated?, Topology Appl. 163, 2-10 (2014). ZBL1285.54018.
PROPOSITION (Alas, Junqueira and Wilson): Let $X$ be a Lindelof $P$-space where every Linearly Lindelof subspace is Lindelof. Then $X$ is weakly discretely generated.
EDIT 14/10/2023: I should mention that we were even lacking an example of a regular $P$-space which is not weakly discretely generated. Such an example has been constructed by Szeptycki and myself in a paper that came out last year:
Spadaro, S.; Szeptycki, S., A regular non-weakly discretely generated (P)-space, Acta Math. Hung. 166, No. 1, 92-96 (2022). ZBL1499.54041.
Unfortunately it is not Lindelof.
