Definition: Heine-Borel Property Defines Compact Subsets 

A subset \(U\) of a metric (or topological) space¹ \(X\) 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

\[U\subset U_{i_1}\cup U_{i_2}\cup \ldots \cup U_{i_n}.\]

Notes

• The finite union $U_{i_1}\cup U_{i_2}\cup \ldots \cup U_{i_n}$ is sometimes referred to as an open subcover.
• The existence of a finite subcover for every open cover of $U$ is called the Heine-Borel property of \(U\) (and is due to Heinrich Eduard Heine and Émile Borel.

¹ The above definition makes no reference to any kind of a metric of the space $X$. Thus, it can be used in more generalized topological spaces rather than metric spaces.

| | | | | created: 2017-02-20 23:43:11 | modified: 2020-06-03 16:58:35 | by: bookofproofs | references: [582]

1.Explanation: Explanation of the Heine-Borel Property