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For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\kappa$ such that for every matching ${\cal M}\subseteq \tau$ we have $|{\cal M}|\leq \kappa$.

Recall that $D\subseteq S$ is dense if $D$ intersects every non-empty open set. In the language of hypergraphs, dense sets are vertex covers. So we define the vertex covering number $t(X,\tau)$ to be the smallest cardinality that a dense set can have.

König's theorem that for all finite bipartite graphs $B$ we have $\nu(B)=\tau(B)$, when using the equivalents of the definitions above for graphs.

Let $(X,\tau)$ be a non-empty Hausdorff space.

Questions.

1) Do we always have $\nu(X,\tau)\leq t(X,\tau)$?

2) Do we always have $\nu(X,\tau)\geq t(X,\tau)$?

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What you are calling the "matching number" of $X$ is usually called its Souslin number -- the smallest cardinal bounding the size of any collection of pairwise disjoint open subsets of $X$.

What you are calling the "vertex covering number" of $X$ is usually called its density.

Thus your question is how the Souslin number and the density of a Hausdorff space are related. The answer is that the density is always $\geq$ the Souslin number, but the inequality can be strict. Thus the answer to your first question is yes, and the answer to the second question is no.

(Denote by $S(X)$ and $d(X)$ the Souslin number and density of $X$, respectively. To see that $S(X) \leq d(X)$, just observe that any dense subset $D$ of $X$ is at least as large as a collection $\mathcal U$ of pairwise disjoint open subsets of $X$, because $D$ must contain a point from each member of $\mathcal U$. To see that this inequality can be strict, try to show that the space $[0,1]^\kappa$ has countable Souslin number, but fails to be separable for any $\kappa > \mathfrak{c}$.)

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  • $\begingroup$ Thanks also for clearing up the terminology! $\endgroup$ – Dominic van der Zypen Jul 30 '18 at 9:58
  • $\begingroup$ For Dominic: for showing that the space $[0, 1]^\kappa$ has Souslin number $\aleph_0$, a useful search term might be "countable chain condition" (ccc). It can be shown that if $X_i$ is a family of spaces such that the product of any finite number of them is ccc, then the product of all of them is ccc. (The hypothesis holds if for example all the $X_i$ are separable.) There's a write-up of this here: ncatlab.org/nlab/show/countable+chain+condition. $\endgroup$ – Todd Trimble Jul 30 '18 at 14:57
  • $\begingroup$ @ToddTrimble Thanks for providing this terminology; I was wondering how the Souslin number and density behave under products, and I think after some searches (and some thinking :-)) I get a clearer picture now $\endgroup$ – Dominic van der Zypen Jul 31 '18 at 7:05
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    $\begingroup$ @DominicvanderZypen One thing that may be surprising in all this is that even if $X$ and $Y$ are ccc, $X \times Y$ need not be. This is connected with independence phenomena in set theory. $\endgroup$ – Todd Trimble Jul 31 '18 at 10:59
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    $\begingroup$ The Souslin number is also called the cellularity of the space and it is often denoted by $c(X)$. What the OP calls "a matching" is also called a cellular family. $\endgroup$ – Ramiro de la Vega Aug 2 '18 at 14:08

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