In mathematics, a compact space is a type of topological space that is both closed and bounded. This means that every open cover of the space has a finite subcover, which is a smaller collection of open sets that still covers the entire space. Compactness is an important property in various areas of analysis and topology. One of the most well-known examples of a compact space is the closed interval [0, 1] in the real numbers. In this interval, every sequence of points has a subsequence that converges to a point within the interval, illustrating the concept of compactness in a tangible way.