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?

A continuous image of a sequentially compact space is sequentially compact. — Mark, Aug 13 '11 at 19:55

score 18 · Answer 1 · answered Aug 13 '11 at 20:05

18

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 $\beta\mathbb N$.

answered Aug 13 '11 at 20:05

Jonas Meyer

53,602

thanks! nice argument too. – user10 Aug 13 '11 at 22:15

score 10 · Answer 2 · answered Aug 13 '11 at 21:07

10

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

answered Aug 13 '11 at 21:07

Henno Brandsma

242,131

score 9 · Accepted Answer · answered Aug 13 '11 at 20:12

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.

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

3 Answers3