Exercise · ZW-COMPACT-SPACE-006 · ★★☆ Intermediate
Let \(X=\{a,b,c,d,e\}\) with the discrete topology and the cover \(\{a,b\},\{c,d\},\{e\},\{b,c\},\{d,e\},\{a,e\}\). What is the least number of sets in a subcover ?
Choose one option.
A \(2\) B \(3\) C \(4\) D \(6\) Show answer ← Back to the question
Explanation Two sets of size at most \(2\) cover at most \(4\) points, but \(X\) has \(5\). Three sets, \(\{a,b\},\{c,d\},\{e\}\), already cover everything.
A topological space \(X\) is compact if every open cover \(\mathcal{U}\) of \(X\) has a finite subcover.
\[X=\bigcup_{U\in\mathcal{U}}U\ \Rightarrow\ X=U_1\cup\dots\cup U_n,\ U_i\in\mathcal{U}\]
More practice problems: compact space