Exercise · ZW-COMPACT-SPACE-007 · ★★☆ Intermediate
Let \(X\) be an infinite set with the cofinite topology (the open sets are \(\varnothing\) and complements of finite sets). Is it compact?
Choose one option.
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- D. Yes: a nonempty open set misses only finitely many points
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Explanation
Given a cover, a nonempty \(U\) omits only finitely many points; cover each by one more open set and the total is finite.
A topological space \(X\) is compact if every open cover \(\mathcal{U}\) of \(X\) has a finite subcover.
\[X=\bigcup_{U\in\mathcal{U}}U\ \Rightarrow\ X=U_1\cup\dots\cup U_n,\ U_i\in\mathcal{U}\]
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