 The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well.In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness.On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point.