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Teichmüller–Tukey lemma

A family of sets ${\displaystyle {\mathcal {F}}}$ is of finite character provided it has the following properties:

1. For each ${\displaystyle A\in {\mathcal {F}}}$, every finite subsetof${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.
2. If every finite subset of a given set ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$, then ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.

Statement of the lemma[edit]

Let ${\displaystyle Z}$ be a set and let ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {P}}($Z)} . If ${\displaystyle {\mathcal {F}}}$ is of finite character and ${\displaystyle X\in {\mathcal {F}}}$, then there is a maximal ${\displaystyle Y\in {\mathcal {F}}}$ (according to the inclusion relation) such that ${\displaystyle X\subseteq Y}$.^[2]

Applications[edit]

Inlinear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider the collection ${\displaystyle {\mathcal {F}}}$oflinearly independent sets of vectors. This is a collection of finite character. Thus, a maximal set exists, which must then span V and be a basis for V.

Notes[edit]

References[edit]

• Brillinger, David R. "John Wilder Tukey" [1]