# A general form of Tukey's Lemma

In the book "General Topology" by John Kelley, Tukey's Lemma is stated as: "If a family of sets is of finite character, it has a maximal member". In the end of the section on set theory, there is the following paragraph:

[...]Finally, it may be noted that, although the formulation of Tukey's Lemma which is given is more or less standard, it does not imply (directly) the most commonly cited applications (for example, each group has a maximal abelian subgroup). There is a more general form which states (very roughly): if a family $\mathfrak{a}$ of sets is defined by a (possibly infinite) number of conditions such that each condition involves only finitely many points, then $\mathfrak{a}$ has a maximal member.

I want a reference to some document in which this general form of Tukey's Lemma is proved and made precise in an easy-to-follow manner. Thanks in advance.

• I'm not sure, but have you seen compactness theorem?
– MJD
Jan 2 '14 at 14:54
• @MJD I am not sure this is what I am looking for. I will look it through more thoroughly though. Jan 2 '14 at 15:02
• @MJD: Amusing. A student of mine wrote to me that in another course the teacher talked about Tukey's lemma and said it is a "weakening of Zorn's lemma", and she didn't understand how. I wrote in return that I see it more of a strengthening of the compactness theorem. :-) Jan 2 '14 at 15:27