Singleboundary

Singleboundary is a term used in topology and related areas to describe a domain whose boundary consists of a single connected component. In this sense, a singleboundary domain has one boundary loop or surface, as opposed to domains with multiple boundary components such as holes.

In the plane, common examples of singleboundary domains include the open disk and the upper half-plane. The unit disk, D = { z : |z| < 1 }, has boundary ∂D = { |z| = 1 }, a single connected set. The upper half-planeH = { z : Im(z) > 0 } has boundary equal to the real line, also a single connected set. By contrast, an annulus { z : r < |z| < R } has two boundary components (the circles |z| = r and |z| = R), and thus is not a singleboundary domain.

The concept is closely related to Jordan domains, which are domains whose boundary is a Jordan curve (a simple closed curve). A Jordan domain is, in particular, a singleboundary domain, and such domains admit conformal maps to the unit disk by the Riemann mapping theorem. However, not every singleboundary domain must have a boundary that is a Jordan curve; the boundary may be connected without being a simple curve.

In practical terms, recognizing a singleboundary domain matters in conformal mapping, potential theory, and PDEs, where boundary conditions and mapping properties often depend on the topology of the boundary. In computational settings, singleboundary shapes are easier to handle for boundary discretization and mesh generation because they involve only one boundary loop.

See also: boundary component, Jordan curve, domain, conformal mapping, Riemann mapping theorem.