Tychonoff's Theorem/General Case/Proof 2/Lemma 2
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Lemma
Let $X$ be a set.
We say that a family $\CC$ of subsets of $X$ has the "finite intersection property" if:
- $\ds \bigcap_{S \in \SS} S \ne \O$
for all finite subsets $\SS \subseteq \CC$.
Let:
- $\ds S = \set {\CC \subseteq \AA : \AA \text { has the finite intersection property} }$
Let $\AA$ be a $\subseteq$-maximal element of $S$ given by Lemma 1.
Let $A, B \in \AA$.
Then $A \cap B \in \AA$.
Proof
Let $\set {C_1, C_2, \ldots, C_n}$ be a finite subset of $\AA$.
Then $\set {A, B, C_1, C_2, \ldots, C_n}$ is a finite subset of $\AA$.
Then since $\AA$ has the finite intersection property:
- $\ds A \cap B \cap \bigcap_{i \mathop = 1}^n C_i = \paren {A \cap B} \cap \bigcap_{i \mathop = 1}^n C_i \in \AA$
for every finite subset $\set {C_1, C_2, \ldots, C_n}$ of $\AA$.
So $\AA \cup \set {A \cap B} \in S$ also has the finite intersection property.
Since $\AA$ is $\subseteq$-maximal in $S$, we have $\AA \cup \set {A \cap B} = \AA$.
That is:
- $A \cap B \in \AA$
$\blacksquare$