What does a connected $F$-space look like? In this MO answer, Joseph Van Name defines an $F$-space as a

*

*complete regular topological space $X$ such that,

*for any function $f\in C(X)$ with zero set $V(f)$, any boundedFn. 1 $g\in C(X\setminus V(f))$ can be extended continuously to all of $X$.

Van Name then goes on to state (along with a number of conditions equivalent to $F$): "There are also some connected $F$-spaces."
Obviously, the one-point space is connected and $F$.  Are there any examples of such spaces with at least two points?
Van Name points out that any infinite compact $F$-space must contain a homeomorphic copy of $\beta\mathbb{N}$, and Nik Weaver's answer to the same question indicates that any convergent sequence in $X$ must be eventually constant.  Conversely, $\beta\mathbb{N}$ is disconnected since $\mathbb{N}$ is.  I don't know how much these constrain things.

Fn. 1 I originally forgot this qualifier, but that changes my definition to the much stronger condition of a $P$-space (thanks, Ulli!).  
 A: As it turns out, answering this question is treated in Gillman, Leonard; Henriksen, Melvin (1956), "Rings of continuous functions in which every finitely generated ideal is principal", Transactions of the American Mathematical Society, 82 (2): 366–391.  The key is to show that any coronal set (of a nice space) is $F$; then they show $X=\beta[0,\infty)\setminus[0,\infty)$ is also connected.  If I'm not mistaken, the proof generalizes to any connected, locally compact space with exactly one end.
Unfortunately, I know of no way to show that $X$ is an $F$-space that is clean, elementary, and correct; rather than read Gillman & Henriksen, I recommend (as suggested by Ulli's answer) Gillman, Rings of Continuous Functions, pp. 210-211, which has a clean proof through a little algebraic geometry.
To see that $X$ is connected, suppose for contradiction otherwise.  Then there exists a surjection $F\in C(X\to\{0,1\})$.  Now, $\beta[0,\infty)$ is normal, since it is compact Hausdorff.  Moreover, $[0,\infty)$ is locally compact, so open in $\beta[0,\infty)$; thus $X$ is closed in the same.  By the Tietze extension theorem, $F$ extends to $G\in C(\beta[0,\infty)\to[0,1])$.  Picking ultrafilters with different values of $F$, there exists $\{x_n\}_n,\{y_n\}_n\to\infty$ such that, for any $n$, we have $G(x_n)=1$, $G(y_n)=0$, and (by passing to a subsequence) $x_n<y_n$.  But then the IVT constructs $z_n\in(x_n,y_n)$ with $G(z_n)=\frac{1}{2}$.  The corresponding ultrafilter $z$ has $F(z)=\frac{1}{2}\notin\{0,1\}$.
A: Let $X$ be a completely regular topological space.
Usually, $X$ is called  F-space, if every cozero set is C*-embedded (i.e. every continuous bounded function to the reals can be extended to the whole space). This is the definition in Gillman, Jerison, Rings of continuuous functions and also the one cited by Joseph van Name in the above mentioned link.
A related condition is
to require that every cozero set is even C-embedded (i.e. every continuous function can be extended). This is, however, equivalent to  P-space (i.e., each $G_\delta$ subset is open), which is much stronger than F-space. For instance, a P-space is strongly zero-dimensional, hence not connected, if infinite. And never compact unless finite.
In contrast to this, $X$ is F-space, if and only if its Stone-Cech compacttification is F-space. Hence there are a lot of compact F-spaces.
Further, if $X$ is locally compact and $\sigma$-compact, then $\beta X \setminus X$ is F-space.
Hence, as you mentioned in your answer, $\beta[0,\infty)\setminus[0,\infty)$ is a compact F-space and also connected.
I'm sure all of these results (and much more) can be found in the great book of Gillman, Jerison, Rings of continuous functions.
