I am looking for a space as in the title, i.e.,

Is there a metacompact, normal, CCC space which is not Lindelof?

A space is ccc iff any family of pairwise disjoint open sets is at most countable.

A space $X$ is metacompact iff for any open cover $\mathcal U$ of $X$ there is a point finite refintement of $\mathcal U$.



1 Answer 1


Yes, such an example can be obtained using the Pixley-Roy hyperspace construction.

Given a topological space $X$, the Pixley-Roy hyperspace on $X$ ($PR(X)$) is defined as the space of all non-empty finite subsets of $X$ with the topology generated by sets of the form $[F,U]:=\{G \in PR(X): F \subset G \subset U \}$, where $F \in PR(X)$ and $U$ is an open subset of $X$ which contains $F$.

It is easy to see that $PR(X)$ is Lindelof if and only if $X$ is countable. Indeed, when $X$ is uncountable, the family $\{[\{x\}, X]: x \in X \}$ provides an open cover without countable subcovers. It's also easy to see that if $X$ has countable weight then $PR(X)$ is ccc.

Eric van Douwen proved that $PR(X)$ is always metacompact (provided $X$ is $T_1$) and that $PR(X)$ is a Moore space when $X$ is first-countable.

So for every uncountable set of reals $Z \subset \mathbb{R}$, the space $PR(Z)$ is not Lindelof, but it's Tychonoff, metacompact, ccc and a Moore space (you didn't ask for the latter, but it's a nice bonus to have! ;-)).

Only missing is normality. To get it we can use a standard trick, originally due to Bob Heath, which goes back to the Normal Moore Space Problem. Let $Y$ be a $Q$-set (that is an uncountable subset of the reals whose every subset is a relative $G_\delta$) with the additional property that $Y^n$ is a $Q$-set, for every $n \in \mathbb{N}$ (the existence of such a set of reals is consistent with and independent from ZFC). Then we can prove that $PR(Y)$ is normal.

For the proofs see the survey Lutzer, David J., Pixley-Roy topology, Topology proceedings, Vol. 3, No.1, Proc. Topol. Conf. Univ. Okla. 1978, 139-158 (1979). ZBL0435.54007.).


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