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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 $|M| = \alpha$.
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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 $2^{2^\alpha}$.)

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  • $\begingroup$ Thanks for your detailled answer, as well as answering the variant for $\alpha < \kappa$! $\endgroup$ – Dominic van der Zypen Nov 22 '19 at 20:04

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