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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 $\mathbb{R}$.

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    $\begingroup$ Using $\omega$ for anything other than the least infinite ordinal, in the context of set theory, is a crime punishable by public flogging. Or at least it should be. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 6, 2015 at 7:43
  • $\begingroup$ @ Asaf Karagila: OK, agreed, but in the context of topology and dimension theory, coverings are often denoted by $\omega$. $\endgroup$
    – Lev Balkanski
    Commented Jan 6, 2015 at 7:49

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It appears in the Hitting Set Problem, and so it would make some sense to call $\tau(\omega)$ the hitting set cardinality.

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    $\begingroup$ Thanks, maybe I'll call it "hitting number", unless there is no conflict with some other similar term. $\endgroup$
    – Lev Balkanski
    Commented Jan 6, 2015 at 9:32

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