Given a topological space $(X,\tau)$, we say that a *matching* is a collection of non-empty open sets that are pairwise disjoint.


Given an infinite cardinal $\kappa$, is there a $T\_2$-space with $|X|\geq\kappa$ and a cardinal $\alpha<|X|$ with the following properties?


1. there is a dense subset $D\subseteq X$ with $|D| = \alpha$, and
2. there is a matching $M\subseteq (\tau\setminus\{\emptyset\})$ with $|M| = \alpha$.



$T_2$-space with a matching equalling the density number
The answer is yes.
Given a cardinal $\kappa$, let $\lambda = 2^\kappa$ (this denotes cardinal exponentiation) and let $X\_0 = 2^\lambda$ (this denotes a Tychonoff product). Clearly $|X\_0| > \kappa$. $X$ has a dense subset of size $\kappa$, and it has the ccc, which means that every "matching" in $X\_0$ is countable. (By the way, what you're calling a *matching* is usually called a *cellular family*.) If we take $X$ to be $\kappa$ disjoint copies of $X\_0$ and set $\alpha = \kappa$, then all your conditions are met.


By the way, an interesting variant of this question is obtained by asking for $\alpha < \kappa$ instead of $\alpha < |X|$. In this version, the answer is yes for some $\kappa$ and no for others. (By the argument above, we get a yes answer if there is some $\alpha$ with $\alpha < \kappa \leq 2^\alpha$. But the answer is no if $\kappa$ is a strong limit cardinal, because if a Hausdorff space has a dense subset of size $\alpha$, then its cardinality is at most $2^{2^\alpha}$.)


