# $T_2$-space with a matching equalling the density number

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 $$|M| = \alpha$$.

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 $$2^{2^\alpha}$$.)
• Thanks for your detailled answer, as well as answering the variant for $\alpha < \kappa$! – Dominic van der Zypen Nov 22 '19 at 20:04