Let $X$ be a compact Hausdorff space. If $X$ is extremally disconnected then the complement of every point $x$ in $X$ is C\*-embedded in $X$ (i.e every continuous bounded real-valued function on $X\setminus\{x\}$ extends to a continuous function on $X$) because every open subset of $X$ is C\*-embedded [Gillman and Jerison, Rings of Continuous Functions; 1H].


If $X$ is basically disconnected then the complement of every non-P-point $x$ is C\*-embedded in $X$ (because $x$ lies in the boundary of a cozero set and cozero sets in basically disconnected spaces are C\*-embedded and have clopen closure [Gillman and Jerison; 1h, 14.25]).


My question is whether the complement of every non-P-point in a compact F-space is C\*-embedded (an F-space is a Hausdorff space in which disjoint cozero sets are contained in disjoint zero sets).


In the opposite direction, I am interested in spaces in which the complement of every non-P-point is C\*-embedded. Can this occur in a non-F-space?



F-spaces and points whose complements are C*-embedded
Consider the $F$-space $\omega^\* $ (the Cech-Stone remainder of $\omega$).
Under the Continuum Hypothesis there is no point whose complement is $C^\* $-embedded. On the other hand it is also consistent that the complement of *every* point of $\omega^\* $ is $C^\* $-embedded. See [this paper](http://dutiaw37.twi.tudelft.nl/~kp/publications/the_papers/open_problems_on_betaN.pdf) for references (page 102).
Furthermore, Alan Dow proved that in the Miller model the complement of a point is $C^\* $-embedded iff the point is not a $P$-point, see [this paper](http://matwbn.icm.edu.pl/ksiazki/fm/fm152/fm15212.pdf).


