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. Compact sets are important because they allow for the application of various mathematical theorems, such as the Heine-Borel theorem, which characterizes compact subsets in Euclidean spaces. In practical terms, compact sets can be visualized as closed intervals on the real number line, like the set a, b where a and b are real numbers. In higher dimensions, a compact set could be a closed ball or a rectangle. Compactness ensures that every open cover of the set has a finite subcover, making it easier to work with in analysis and other areas of mathematics.