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For $X$ a countable $T_{3_{1/2}}$ space then the Stone-Cech compatification $\beta(X)$ has size $2^{2^{\aleph_0}}$.

Also Shelah has written things about Dowker spaces of size $\aleph_{\omega+1}$.

Still in topology, $\mathbb{N}^{\aleph_1}$ is not a Cech-complete space. Recall that a Cech-complete space is space where the remainder $\beta(X) \backslash X$ is a $G_\delta$ set.

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For $X$ a $T_{3_{1/2}}$ space then the Stone-Cech compatification $\beta(X)$ has size $2^{2^{\aleph_0}}$.

Also Shelah has written things about Dowker spaces of size $\aleph_{\omega+1}$.

Still in topology, $\mathbb{N}^{\aleph_1}$ is not a Cech-complete space. Recall that a Cech-complete space is space where the remainder $\beta(X) \backslash X$ is a $G_\delta$ set.

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For $X$ a $T_{3_{1/2}}$ space then the Stone-Cech compatification $\beta(X)$ has size $2^{2^{\aleph_0}}$.

Also Shelah has written things about Dowker spaces of size $\aleph_{\omega+1}$.

Still in topology, $\mathbb{N}^{\aleph_1}$ is a Cech-complete space. Recall that a Cech-complete space is space where the remainder $\beta(X) \backslash X$ is a $G_\delta$ set.