Question

A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open .$($i.e $\tau$ is closed under countable intersection$)$. Here we recall some special properties of $P$-spaces:

• Every countable subset of $X$ is obviously closed and discrete.

• Every countable subset of $X$ is $C$-embedded in $X$.$($i.e. every continuous real valued function on a countable subset of $X$ can be extended to all of $X$ $)$

Now with the sake of above properties I could pose my Questions. My questions that are given as follows are the extended form of these properties of countable sets to Lindelöf subsets of $P$-spaces.

• Is it true that in every $P$-space, every Lindelöf subset is closed?

• Is it true that in every $P$-space every Lindelöf subset is $C$-embedded in $X$?

Answer 1

Every Lindelof subset of a $P$-space is closed, and the proof is almost the same as the proof of "a compact subset of a Hausdorff space is closed" (I´m assuming your space is Hausdorff since you wrote that every countable set is obviously closed).

I´m not so sure about the second question, but every $P$-space is an $F$-space and every Lindelof subset of an $F$-space is $C^*$-embedded. You could take a look at Negrepontis´ article "On the product of $F$-spaces" for a proof of this fact and try to adapt it to your situation.