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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 $X$?

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  • $\begingroup$ Dear @Hachino, please remember to not edit more than three old questions each day. (An old question here is one not already on the front page.) Thank you. You may read more at meta.mathoverflow.net/questions/599/… $\endgroup$
    – Ricardo Andrade
    Commented Apr 24, 2015 at 20:31
  • $\begingroup$ @RicardoAndrade : I hear and will comply, sorry for yesterday's burst. This shall not happen anymore. $\endgroup$
    – Hachino
    Commented Apr 25, 2015 at 8:33
  • $\begingroup$ @Hachino, no problem. It takes a while to figure out some of these "rules". Good luck! $\endgroup$
    – Ricardo Andrade
    Commented Apr 25, 2015 at 9:38

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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.

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  • $\begingroup$ Hello Dear Ramiro. At first I have to say that I am so sorry about my Delay. Thank you very much for your refrence and guidance .When I posed these problems, I wanted to improve exercise [3B] of the text gillman-jerison. For the second question, I think you were very closed to show it. It suffices to apply the following theorem for $C$-embedded subsets. Theorem:A $C^*$-embedded subset of topological space $X$ is $C$-embedded iff it is completely separated from every zero-set which is disjoint from it. As you Know in this case every zero-set is clopen and that's all. Thank's a lot. $\endgroup$
    – Ali Reza
    Commented May 31, 2012 at 18:00

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