= Finite intersection characterization of compactness
{title2=$\{F_i\}\text{ closed with the finite intersection property}\ \Longrightarrow\ \bigcap_iF_i\ne\varnothing$}
A <topological space> is <compact> exactly when every family of <closed sets> with the <finite intersection property> has nonempty <intersection>. Take <complements> and apply <De Morgan's laws>: an empty total <intersection> corresponds to an <open cover>, while an empty finite <intersection> corresponds to a <finite subcover>. This works without any <Hausdorff condition>. With the convention that an empty-family <intersection> is the whole ambient <set>, the <finite intersection property> includes nonemptiness of that ambient <set>, and the criterion correctly treats the empty <compact space>.
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