2
$\begingroup$

I'm studying the ordinal space $[0,\kappa[$ where $\kappa\neq \omega$ is a cardinal of countable cofinality and I want to know why there are in $[0,\kappa[$ two disjoint closed sets of cardinality $\kappa$.

The case when $\kappa=\omega$ it's obvious but I can't prove the general one... If anybody could help me I would be more than gratefull.

Thanks!

Improve this question
$\endgroup$
4
  • $\begingroup$ $\kappa=\omega+1$ is not a cardinal number... $\endgroup$
    – Ergonvi
    Commented Jan 5, 2015 at 10:04
  • 1
    $\begingroup$ Didn't you ask that on math.SE before? $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 5, 2015 at 10:04
  • $\begingroup$ @Ergonvi -- sorry, I removed my wrong comment before I saw your correction (you were fast! :-). I overlooked the phrase countable cofinality. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 5, 2015 at 10:07
  • $\begingroup$ @Ergonvi -- and yes, I overlooked wordcardinality too. I had overlooked everything! $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 5, 2015 at 10:12

1 Answer 1

Reset to default
3
$\begingroup$

Let $\kappa_n$ be increasing cofinal in $\kappa,$ and let

$X=\bigcup_{n}(\kappa_{2n}, \kappa_{2n+1}]$, and $Y=\bigcup_n (\kappa_{2n+1}, \kappa_{2n+2}]$

Improve this answer
$\endgroup$
5
  • $\begingroup$ Thank you @Mohammad Golshani ! But why $X$ and $Y$ have cardinality $\kappa$? $\endgroup$
    – Ergonvi
    Commented Jan 5, 2015 at 10:03
  • $\begingroup$ That's trivial, for example $|X|=\Sigma_n \kappa_{2n+1}=\kappa.$ Similarly $|Y|=\Sigma_n \kappa_{2n+2}=\kappa.$ $\endgroup$
    – Mohammad Golshani
    Commented Jan 5, 2015 at 10:25
  • $\begingroup$ Oh, that's right... Thank you @Mohammad Golshani! :) $\endgroup$
    – Ergonvi
    Commented Jan 5, 2015 at 17:59
  • $\begingroup$ But the same argument wouldn't be right for cardinals with uncontable cofinality? I mean for example, $X=\bigcup_{\alpha <\omega_1} (\kappa_{2\alpha},\kappa_{2\alpha+1}]$ and $Y=\bigcup_{\alpha <\omega_1} (\kappa_{2\alpha+1},\kappa_{2\alpha}]$... $\endgroup$
    – Ergonvi
    Commented Jan 16, 2015 at 20:36
  • $\begingroup$ It doesn't work,for example then $\kappa_{\omega}$ should be in both of $X$ and $Y$. $\endgroup$
    – Mohammad Golshani
    Commented Jan 18, 2015 at 4:43

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .