continuum

A continuum is a concept used in several contexts to denote a notion of an unbroken, continuous whole. In everyday language it refers to something that can be extended without gaps. In mathematics, the idea is made precise in a few related ways, often focusing on the absence of discrete steps or gaps.

In topology, a continuum is defined as a nonempty compact connected metric space. The closed interval [0,1] is a standard example. By this definition the real line R is connected but not compact, and thus not a continuum in the strict topological sense, though it is frequently described as a continuum in a looser sense. The notion highlights properties such as path-connectedness and the ability to vary points continuously.

In set theory, “the continuum” denotes the cardinality of the real numbers, usually written c or 2^aleph-null. This emphasizes the size of the continuum rather than its topological shape. A central question about this size is the continuum hypothesis (CH), which posits that there is no set whose cardinality lies strictly between that of the integers and the real numbers. CH is independent of the standard axioms of set theory (ZFC): both CH and its negation can be consistently added to ZFC if ZFC itself is consistent.

In science and applied mathematics, the continuum concept underpins modeling in continuum mechanics and field theories, where matter and space are treated as continuous media rather than as discrete collections. Related ideas include continuity of functions in calculus, which describes smooth variation without jumps.

Etymology traces continuum to Latin continere, meaning to hold together, reflecting the idea of an unbroken whole.