A compact set is a fundamental concept in mathematics, particularly in topology. It refers to a set that is both closed and bounded. This means that the set contains all its limit points and fits within a finite space. For example, in the real number line, a closed interval like a, b is compact because it includes its endpoints and does not extend infinitely. In more general terms, compactness ensures that every open cover of the set has a finite subcover. This property is crucial in various areas of analysis and geometry, as it allows for the application of important theorems, such as the Heine-Borel theorem in Euclidean spaces.