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An ultrafilter that contains the cofinite filter is called the free ultrafilter. Why is it called free? Does this notion of freeness bear any relation to the way we say the free vector space on a set?

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There are several words which are used in completely orthogonal meanings through mathematics. Free, normal, regular, etc. it doesn't mean that there is any relation between those. – Asaf Karagila Sep 15 '12 at 9:18
up vote 8 down vote accepted

In this context free contrasts with fixed: an ultrafilter $\mathscr{U}$ on a set $X$ is said to be fixed if there is an $x\in X$ such that $\mathscr{U}=\{U\subseteq X:x\in U\}$. In other words, this $\mathscr{U}$ is just the family of all subsets of $X$ containing the point $x$: it’s tied down, or fixed, to the point $x$. (Such ultrafilters are also known as principal ultrafilters.) An ultrafilter that is not fixed in this way $-$ that is not so tied down $-$ is free. Its members are in some sense free to ‘float’ all over $X$: they’re not forced to contain any particlar element.

This notion has nothing to do with free in the sense that you mention, or in the sense of a free group.

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