Exercise · ZW-COMPACT-SPACE-010 · ★★★ Advanced
Let \(X=\mathbb{N}\) with open sets \(\varnothing\) and \(U_k=\{k,k+1,\dots\}\), \(k\ge1\). Is it compact?
Choose one option.
Show answer
- A. Yes: every open set containing \(1\) equals \(X\)
Explanation
A cover must include an open set containing \(1\), and the only one is \(U_1=X\). That single set is already a finite subcover.
Review the concept: Compact space
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
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