continuum

In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. Definitions A continuum that contains more than one point is called nondegenerate. A subset A of a continuum X such that A itself is a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum. A continuum X is homogeneous if for every two points x and y in X, there exists a homeomorphism h: X → X such that h(x) = y. A Peano continuum is a continuum that is locally connected at each point. An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum X is hereditarily indecomposable if every subcontinuum of X is indecomposable. The dimension of a continuum usually means its topological dimension. A one-dimensional continuum is often called a curve. Examples