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.

Let $\lambda<\omega_1$ be a limit ordinal and consider the set $X=\{\alpha+1\colon \alpha<\lambda\mbox{ is a limit ordinal}\}$ Is the characteristic function $\chi_X$ continuous on $\omega_1$ with interval topology?

share|improve this question
add comment

1 Answer 1

Notice that $\omega \cdot \omega = \sup_{n\in\omega} \omega \cdot n = \sup_{n\in\omega} (\omega\cdot n+1)$.

This implies that in the interval topology $\lim_{n\to\infty} (\omega \cdot n+1)= \omega\cdot\omega$. But $1=\lim_{n\to\infty} f(\omega\cdot n+1)\ne f(\omega\cdot\omega)=0$, and thus this function is not continuous.


I assume that by interval topology you mean the topology generated by the subbase consisting of intervals $\{\xi; \xi<\beta\}$ and $\{\xi; \beta<\xi<\omega_1\}$ for $\beta<\omega_1$, see e.g Komjath, Totik: Problems and theorems in classical set theory p.40. Note that this book also contains an exercise containing some characterization of functions which are continuous with respect to interval topology.

You can find out more about this book on the website of one of the authors.

share|improve this answer
    
Martin: in the second limit equation you forgot $f$. Why do you use \ldot instead of \cdot? –  Damian Sobota Nov 29 '11 at 23:18
    
@Damian Thanks for noticing the forgotten $f$. I am used to write $a.b$ or $ab$ instead of $a\cdot b$. I believe it is not that unusual. According to wikipedia Multiplication is sometimes denoted by either a middle dot or a period: $5\cdot 2$ or $5.2$. Although of course my notation is confusing for people from countries where period is used as decimal separator. –  Martin Sleziak Nov 30 '11 at 12:19
    
In Britain they prefer $a.b$ for multiplication and $3\cdot 14159$ for decimals. The copy editor for the journal will even change your manuscript for you to their system! –  GEdgar Nov 30 '11 at 15:14
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.