# Is there a linearly Lindelöf non-Lindelöf $P$-space?

A completely regular topological space $$(X,\tau)$$ is a $$P$$-space, if every $$G_\delta$$-subset of $$X$$ is open (i.e $$\tau$$ is closed under countable intersections).

A topological space $$X$$ is linearly Lindelöf if every open cover of $$X$$ which is linearly ordered by the subset relation has a countable subcover.

Is there a linearly Lindelöf non-Lindelöf $$P$$-space?

• math.stackexchange.com/questions/474376/… Please, do not double post your question on MSE and on MO. If you see that your question is still without an answer on one of these sites after a while, you can still post it on the other site. Commented Aug 23, 2013 at 15:02
• I don't know why people are voting to close this question. Perhaps at least one of the voters could give a reason? Commented Aug 24, 2013 at 0:36
• Robson: In Arhangelski-Tkachenko's book, the following is listed as an open problem: "is every linearly Lindelof P-group Lindelof?". I've also seen "Is the product of two linearly Lindelof P-spaces linearly Lindelof?" listed as open, so my guess is that your question is also open, but I'm not sure. Commented Aug 24, 2013 at 0:41
• Is it at least true that every discretely Lindelof P-space is Lindelof? "Discretely Lindelof" means "the closure of every discrete set is Lindelof". It's not too hard to see that every discretely Lindelof space is linearly Lindelof. Commented Nov 8, 2017 at 18:23

It was shown by A.K. Misra in "A topological view of $P$-spaces" that any regular $\aleph_1$-Lindelöf $P$-space is normal (see Corollary 4.6). Also, it is not hard to see that any linearly Lindelöf space must be $\aleph_1$-Lindelöf.
Edit: As pointed out by Robson (see comment below), the notion of $\aleph_1$-Lindelöf used by Misra in his paper doesn't correspond to modern terminology. So I don´t know if a regular linearly Lindelöf $P$-space must be normal. However the following is question 486 in "Open problems in topology II":
(Arhangel´ski) Is the product of two linearly Lindelöf $P$-spaces Lindelöf?
So I still think that the OP´s question is open, since product of two Lindelöf $P$-spaces is Lindelöf (so it is the same question after all).
• Corolllary 4.6: Every regular $\aleph_1$-Lindelöf $P$-space is paracompact, hence normal. By the context, I think "$\aleph_1$-Lindelöf" in this corollary means "$L(X)\leq\aleph_1$". Note that if the corollary 4.6 was true for current definition of "$\aleph_1$-Lindelöf" then the question would be answered because: linearly Lindelöf + regular + paracompact $\Rightarrow$ Lindelöf Commented Aug 26, 2013 at 17:59