Timeline for union of Stone-Cech remainders

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Apr 14, 2012 at 7:31	comment	added	Douglas Somerset		Since MathOverflow has been kind enough to re-promote this question, let me raise another issue. In the question I mentioned a dense subalgebra of the C-algebra of continuous bounded function on this space (i.e continuous bounded functions on cofinite subsets of $X$ - see also KPH's answer). What do the rest of the functions in the C-algebra look like? They must be continuous except on a co-countable subset $D$ of $X$, but what can $D$ look like? Specifically, if $X=[0,1]$ can $D$ be the set of rational numbers in the $X$?
Mar 29, 2012 at 17:23	comment	added	Douglas Somerset		Thanks for this. It looks a more natural way of describing the space from a topological viewpoint than my attempt. I take it from the general silence that people have not considered this space before, or at least have not found anything interesting to say about it. One friend that I consulted observed that if $X=[0,1]$ then $Y_{\infty}$ is not an F-space (in spite of having many subspaces which are F-sapces). This is because every open set of $Y_{\infty}$ has a dense cozero set and thus $Y_{\infty}$ would be extremally disconnected and hence would have to coincide with the absolute of $X$.
Mar 29, 2012 at 8:46	history	answered	KP Hart	CC BY-SA 3.0