MathDB

Let

\mathcal{F}

be a nonempty family of sets with the following properties: (a) If

X \in \mathcal{F}

, then there are some

Y \in \mathcal{F}

and

Z \in \mathcal{F}

such that

Y \cap Z =\emptyset

and

Y \cup Z=X

. (b) If

X \in \mathcal{F}

, and

Y \cup Z =X , Y \cap Z=\emptyset

, then either

Y \in \mathcal{F}

Z \in \mathcal{F}

. Show that there is a decreasing sequence

X_0 \supseteq X_1 \supseteq X_2 \supseteq ...

of sets

X_n \in \mathcal{F}

such that

\bigcap_{n=0}^{\infty} X_n= \emptyset.

F. Galvin

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