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



> 
> **Is there a metacompact, normal, CCC space which is not Lindelof?**
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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 $\mathcal U$.


Thanks. 



Is there a metacompact, normal, CCC space which is not Lindelof
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](https://zbmath.org/?q=an:0435.54007).).


