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 $\nu(X,\tau)\geq t(X,\tau)$?