A compact space is a type of mathematical space in topology that is both closed and bounded. This means that it contains all its limit points and fits within a finite region. Compact spaces are important because they allow for certain properties, such as every open cover having a finite subcover, which simplifies many problems in analysis and geometry. One common example of a compact space is the closed interval [0, 1] in the real numbers. In contrast, an open interval like (0, 1) is not compact because it does not include its endpoints. Compactness is a key concept in various areas of mathematics, including functional analysis and differential geometry.