If $X$ is Hausdorff and $|X|> \mathfrak{c}$, does $X$ always have a uncountable discrete subspace? Let $X$ be a Hausdorff topological space with $|X|> \mathfrak c$. Does $X$ always have a uncountable discrete subspace? Thanks for your help.
 A: 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.)

