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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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1 Answer 1

up vote 1 down vote accepted

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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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. –  Ali Reza May 31 '12 at 18:00

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