A subset \(U\) of a metric (or topological) space^{1} \(X\) is called compact, if for every open cover $(U_i)_{i\in I}$ of $U$ there exist only finitely many indices \(i_1,i_2,\ldots,i_k\in I\) with
\[U\subset U_{i_1}\cup U_{i_2}\cup \ldots \cup U_{i_n}\].
Explanations: 1
Chapters: 1
Corollaries: 2 3
Explanations: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23 24 25 26
Theorems: 27 28 29 30
The above definition makes no reference to any kind of metric of the space $X$. Thus, it can be used in more generalized topological spaces rather than metric spaces. ↩