continua

Continua, in topology, are nonempty compact connected metric spaces. The term is commonly used for compact, connected subsets of a metric space; in Euclidean spaces this means compact, connected subsets of R^n. The study of continua and their subspaces is known as continuum theory.

Examples include the closed interval [0,1], the circle S^1, and any convex compact subset of R^n, such as disks and solid polygons. More intricate examples arise in plane topology, such as the Warsaw circle or the pseudo-arc, which illustrate that continua can fail to be locally connected or decomposable. An arc is a continuum homeomorphic to [0,1], while a simple closed curve is a continuum homeomorphic to S^1.

Key properties and concepts: Every subcontinuum of a continuum is itself a continuum. A continuum is decomposable if it can be expressed as the union of two proper subcontinua; otherwise it is indecomposable. The study often distinguishes simple continua like arcs and circles from more complex ones such as indecomposable continua. For a given compact metric space, the collection of its subcontinua forms a compact space under the appropriate Vietoris or Hausdorff-type topology.

Applications and scope: Continuum theory investigates the structure and classification of continua, the relationships among their subcontinua, and the behavior of maps between continua. It intersects with plane topology, geometric topology, and dynamical systems, providing a framework for understanding how connected, compact shapes can be decomposed, approximated, or transformed.