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A topological space $X$ is an $F$-space, if Every finitely generated ideal in the ring of all continuous functions on $X$,denoted by $C(X)$, is principal. The text "Rings of continuous functions" written by Gillman and Jerison , has numerous equivalent conditions of these topological spaces, which some of them are topologic.(see pages 205-215 of the mentioned refrence). Here is an important topological equivalence of these spaces:

Def: A subspace $A$ of topological space $X$ is $C^*$-embedded if every bounded continuous real valued functions on $A$, can be extended to all of $X$.

Theorem:A topological space $X$ is an $F$-space iff every cozero-set$\($i.e. $X-Z(f)$ for some $f\in C(X)$$\)$ is $C^*$-embedded in $X$.

The mentioned refrence, tells us that every $C^*$-embedded subset of an $F$-space is an $F$-space.since cozero-sets and countable subsets of these spaces are $C^ *$-embedded and then $F$-space.

With the above summary I can give my questions.

Q1. The refrence didn't mention to an $F$-space that has a subspace which is not $F$-space. is this space very simple to find?

Q2. Is there a topological space $X$ with the property that every countable subspace is $C^ *$-embedded, but $X$ not an $F$-space?

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A random question, pardon my ignorance: What is the relationship between these spaces and the $F$-spaces in functional analysis, if any? – Christopher A. Wong May 22 '12 at 21:42

A completely regular space is an $F'$-space if disjoint cozero sets have disjoint closure. Every $F$-space is an $F'$-space and every normal $F'$-space is an $F$-space. There is an example (1.10) in Alan Dow, 'On $F$-spaces and $F'$-spaces', Pac. J. Math., 108 (1983) 275-284 of a locally compact $F'$-space which is not an $F$-space but which is an open subset of a compact $F$-space.

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