👉 In mathematics, a pre-compact set is a subset of a metric space that is closed and bounded but not compact. A compact set is a set that contains all its limit points. Pre-compact sets are particularly important in topology and analysis because they can be used to prove theorems about open and closed subsets of metric spaces. A pre-compact set is defined as follows: 1. The set S is said to be pre-compact if it is both closed and bounded. 2.