Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
3  
A continuous image of a sequentially compact space is sequentially compact. –  Mark Aug 13 '11 at 19:55
add comment

3 Answers

up vote 7 down vote accepted

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.

share|improve this answer
add comment

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 $\beta\mathbb N$.

share|improve this answer
    
thanks! nice argument too. –  user10 Aug 13 '11 at 22:15
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.