A compact set is a fundamental concept in topology and mathematics that refers to a set that is both closed and bounded. This means that all its points are contained within a finite distance, and it includes all its limit points. Compact sets are important because they allow for the application of various theorems, such as the Heine-Borel theorem, which states that in Euclidean space, a set is compact if and only if it is closed and bounded. In practical terms, compact sets can be thought of as "small" sets that do not stretch out infinitely. For example, a closed interval like [0, 1] on the real number line is compact, while an open interval like (0, 1) is not, as it does not include its endpoints. Compactness is a key property in many areas of analysis and helps ensure that certain functions behave nicely, such as being continuous or attaining maximum and minimum values.