Franklin's Notes


Finite intersection property

Given a boolean algebra $B$, we say that $B$ has the finite intersection property if and only if the meet (or intersection, or product) of any finite set of elements of $B$ is $\neq 0$ (nonempty). Similarly, we say that a subset $X\subset B$ has the finite intersection property iff the meet of any finite set of elements of $X$ is nonempty. Notice that if we have a chain of subsets each of which possesses the finite intersection property, then their union also has the finite intersection property, for any finite subset $S\subset X$ must be fully contained in some $X_i$, which is known to have the finite intersection property. (This fact is useful in the construction of nonprincipal ultrafilters .)

set-theory

model-theory

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