Question

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?

Answer 1

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 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.