Stone Cech compactification of the naturals a cts image of the Cantor set?

Question

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

Answer 1

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.

Answer 2

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 $\beta\mathbb N$.

Answer 3

A Hausdorff space is the continuous image of the Cantor set iff it is compact metrisable.