Let $X$ be a set and $\omega$ be a family of its subsets. Consider the family
$\mathcal{F}$ of subsets of $X$, such that any $A\in\mathcal{F}$ has a
non-empty intersection with each element of $\omega$. Let $\tau(\omega)$
denote the following cardinal


$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tau(\omega
)=\min\{\mathtt{card}A|~A\in\mathcal{F}\}$.


My question is: Is there a name for the cardinal number $\tau(\omega)$?


I call it temporarily "width" of $\omega$. If $X$ is finite, maybe there is a
term coming from combinatorics. In fact, I am interested in the case when
$\omega$ is a finite family of intervals in $\mathbb{R}$.



Is there a name for this cardinal?
It appears in the Hitting Set Problem, and so it would make some sense to call $\tau(\omega)$ the [hitting set](http://www.nada.kth.se/~viggo/wwwcompendium/node149.html) cardinality.


