Compact Spaces

A compact space in mathematics, particularly in topology, is a set that is both closed and bounded. This means that it is limited in size (bounded) and contains all its edge points (closed). Compact spaces have a crucial property: any open cover—a collection of open sets that together cover the space—always has a finite subcover, meaning that you can select a limited number of those open sets to still cover the entire space. This concept is essential in various areas of mathematics, as it helps ensure certain desirable properties and behaviors in analysis and geometry.