4
$\begingroup$

Let $X$ be a Hausdorff topological space with $|X|> \mathfrak c$. Does $X$ always have a uncountable discrete subspace? Thanks for your help.

$\endgroup$
  • 1
    $\begingroup$ I have downvoted because this question does not meet the quality standards that many users desire for questions on this site: it does not have any background information on where you encountered the problem, nor any information on what you have tried. $\endgroup$ – Carl Mummert May 1 '13 at 19:06
7
$\begingroup$

A result of Hajnal and Juhasz says the following:

Given any Hausdorff space $X$, $|X| \leq 2^{2^{s(X)}}$; in particular every Hausdorff space with countable spread has size $\leq 2^{2^\omega}$.

However, Todorcevic gave the following consistency result:

Assuming PFA, every Hausdorff space with countable spread has size $\leq 2^\omega$.

For the flip side we get the following:

It is consistent that there is a Hausdorff (even collectionwise normal) space with countable spread of size $2^{\omega_1}$.

(This is obtained by adding $\omega_1$ many Cohen reals to a model of CH which additionally satisfies that there is a family of $2^{\omega_1}$ many uncountable subsets of $\omega_1$ such that the intersection of any two distinct members is countable; note that CH will still hold in the extension.)

See the following references for more details:

  • R. Hodel, Cardinal functions I, in the Handbook of Set-Theoretic Topology, pp.1-61. (In particular section 5.)
  • I. Juhasz, Cardinal functions II, in the Handbook of Set-Theoretic Topology, pp.63-109. (In particular section 2.)
| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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