- What does it mean for a topological space to be compact?
Solution
1. A topological space \(X\) is compact if every open cover of \(X\) has a finite subcover.
Explicitly, whenever \(X=\bigcup_{i\in I}U_i\) with each \(U_i\) open, there exist \(i_1,\dots,i_r\in I\) such that \(X=U_{i_1}\cup\cdots\cup U_{i_r}\). This is the definition of compactness.