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After the Russian mathematician I. Parovičenko, a compact Hausdorff space $X$ is called a Parovičenko space iff $X$ satisfies the following conditions: 1) $X$ is zero dimensional and has no isolated points, 2) $w(X)=2^{\aleph_0}$, 3) every non-empty $G_{\delta}$ subset of $X$ has non-empty interior, 4) the closures of any two disjoint zero sets of $X$ are also disjoint. The most well-known Parovičenko space is evidently the growth space $\mathbb{N}^*=\beta\mathbb{N}-\mathbb{N}$. In 1963 Parovičenko has proved the following astonishing fact: in the Set Theory Model $\text{ZFC}+(\aleph_1=2^{\aleph_0})$, all Parovicenko spaces are homeomorphic to $\mathbb{N}^*$ (see \emph{Soviet Math. Doklady} {\bf 4} (1963), 592$-$595). Fifteen years later, E. van Douwen and J. van Mill, two young Dutch topologists, have proved the converse: i.e., if all Parovičenko spaces are homeomorphic (or equivalently, if there is no Parovičenko space other than $\mathbb{N}^*$), then the Continuum Hypothesis $\aleph_1=2^{\aleph_0}$ is true (see Proc. Amer. Math. Soc., 72 (1978), no. 3, 539$-$541). On the other hand, Canadian Murray Bell proved in 1990 that, there is a compact first countable Hausdorff space, which is not a continuous image of $\mathbb{N}^*$ in the model $\text{ZFC}+(\aleph_1<2^{\aleph_0})$ (see Topology Appl. 35 (1990), no. 2-3, 153$-$156).
We will give an explicit proof of van Douwen and van Mill's outstanding result in this talk.