Komplekssete

Komplekssete is a term used in Danish-language mathematics to denote a subset of the complex numbers, typically viewed as a subset of the complex plane C. In this sense, a kompleksset consists of elements z = x + iy with x and y real, and is studied using the standard Euclidean topology induced by C. Basic properties of komplekssets include openness, closedness, boundedness, compactness, and various connectivity notions such as connectedness and path-connectedness. The boundary of a kompleksset comprises points that are limit points of both the set and its complement.

Common examples include simple regions such as the open unit disk { z in C : |z| < 1 }, its closure { |z| ≤ 1 }, half-planes, and strips. More intricate komplekssets arise in complex dynamics and fractal geometry, notably Julia sets and the Mandelbrot set, which are defined as particular subsets of C via iterative processes of polynomials.

Operations on komplekssets follow standard set operations: unions, intersections and complements yield new komplekssets. In complex analysis, holomorphic maps and conformal equivalences are often studied with respect to how they transform komplekssets, for example by preserving openness under preimages or by mapping boundaries in informative ways.

In practice, the concept provides a language for talking about regions and collections of complex numbers, with applications in complex analysis, potential theory, geometric function theory, and dynamical systems. See also complex plane, complex analysis, Julia set, Mandelbrot set, topology of the complex plane.