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Is $\beta \mathbb{N}$, the Stone-Cech compactification of $\mathbb{N}$, a continuous image of the Cantor set?

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    $\begingroup$ A continuous image of a sequentially compact space is sequentially compact. $\endgroup$ – Mark Aug 13 '11 at 19:55
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The Cantor set is sequentially compact, since compactness is equivalent to sequential compactness for metrizable spaces. If $\beta \mathbb{N}$ were a continuous image of it, then it would be sequentially compact too (an easy exercise in point-set topology), but as is quite well known, this is not the case.

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The Cantor set has cardinality $\mathfrak{c}$, and the Stone–Čech compactification of the integers has cardinality $2^{\mathfrak{c}}$. Hence there is no surjective map from the Cantor set to $\beta\mathbb N$.

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  • $\begingroup$ thanks! nice argument too. $\endgroup$ – user10 Aug 13 '11 at 22:15
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A Hausdorff space is the continuous image of the Cantor set iff it is compact metrisable.

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