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In 1958 Gleason [1] constructed projective covers in the category of compact Hausdorff spaces. These may be characterized in many ways. One description that is most interesting to me: $p:EX\to X$ is a projective cover of the compact space $X$ if for any surjection $f:P\to X$ with $P$ compact there exists the oblique arrow $\bar{p}$ as below: $$ \begin{array}{lll} && P \\ & \stackrel{\bar{p}}{\nearrow} & \downarrow \scriptstyle{f} \\ EX & \stackrel{p}{\longrightarrow} & X \end{array} $$ Additionally we require that if $P=EX$ and $f=p$ then $\bar p$ is a homeomorphism.

In particular every such cover $EX$ is a retract of $\beta D$ -- the Stone-Cech compactification of some discrete space $D$.

Question: Do we have projective covers in the category of realcompact spaces?

Of course this question is interesting only in the presence of measurable cardinals.


The Gleason's idea is so beautiful that it begs for generalizations, and indeed I found plenty of related literature, let me mention only a monograph [2]. However all the literature I found restricts the maps $f$ to those for which, roughly, $f^{-1}(x)$ is compact for every $x\in X$. I found only one, failed, attempt to answer the question.

Let me note that the Gleason's construction of $p$ easily generalizes, the problem is in proving the existence of $\bar{p}$ when $f^{-1}(x)$'s are far from compact. A candidate for such $p:E_\nu X\rightarrow X$ may look like this: First we ensure that $X$ has sufficiently many open subsets - let $X^\delta$ be the set $X$ with discrete topology. Let $\pi:\nu X^\delta\rightarrow X$ be the unique extension of the identity to the realcompactification of $X^\delta$. Let $X^*$ be the set $X$ with the quotient topology induced by $\pi$. Now we build the $E_\nu X$ - let $R(X^*)$ be the Boolean algebra of regular closed subsets of $X^*$, that is $C\subseteq X^*$ such that $C=\overline{\mathop{\rm int} C}$. The Stone space of ultrafilters on $R(X^*)$ is called the absolute of $X^*$ and denoted $EX^*$. It seems that our $E_\nu X$ should be the subspace of countably complete ultrafilters in $EX^*$, but I don't see how to obtain the existence of $\bar p$.

[1] A. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958) 482–489.

[2] J. Porter and R. Woods, Extensions and absolutes of Hausdorff spaces. Springer-Verlag, New York, (1988) xiv+856 pp

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