In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.
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👉 Connected space in the context of Polyomino
A polyomino is a plane geometric figure, connected, formed by joining a finite number of unit squares edge-to-edge. It is a polyform whose cells are squares. It may be regarded as a finite and connected subset of the regular square tiling.
Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces made of 1 to 6 squares were first published in Fairy Chess Review between the years 1937 and 1957, under the name of "dissection problems." The name polyomino was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 "Mathematical Games" column in Scientific American.
In this Dossier
Connected space in the context of Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.
Connected space in the context of Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. Thus, an algebraic surface is a solution of a set of polynomial equations, in which there are two independent directions at every point. An example of an algebraic surface is the sphere, which is determined by the single polynomial equation Studying the intrinsic geometry of algebraic surfaces is a central topic in algebraic geometry. The theory is much more complicated than for algebraic curves (one-dimensional cases), and was developed substantially by the Italian school of algebraic geometry in the late 19th and early 20th centuries.It remains an active field of research.
In the simplest cases, algebraic surfaces are studied as algebraic varieties over the complex numbers. For example, the familiar sphere (for real ), becomes a complex (affine) quadric surface, which simultaneously incorporates the sphere and hyperboloids of one and two sheets, and this allows some complications (such as the topology: whether the surface is connected, or simply connected) to be deferred somewhat. Higher degree surfaces include, for example, the Kummer surface. The classification of algebraic surfaces is much more intricate than the classification of algebraic curves, which have dimension one, and is already quite complicated.
Connected space in the context of Fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.