Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be a normal space, If we have a countable local base at the point m in the remainder of Stone -cech compactification $\beta X\setminus X$, can we say the point m has also a countable local base in $\beta X$?

share|cite|improve this question
up vote 1 down vote accepted

No. The Čech-Stone compactification of $\omega_1$ is $\omega_1+1$, its one-point compactification, so the remainder is a singleton, $\{\omega_1\}$. However, the character of the point $\omega_1$ in $\omega_1+1$ is $\omega_1$; it does not have a countable local base. The space $\omega_1$ has the topology induced by a linear order, so it is normal.

share|cite|improve this answer
thanks so much.. – ege Nov 27 '12 at 9:55
@ege: You’re welcome. – Brian M. Scott Nov 28 '12 at 3:16

Your Answer


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.