Connected space

A topological space is connected if it cannot be expressed as the union of two disjoint, nonempty open subsets. Equivalently, its only subsets that are both open and closed are itself and the empty set.

Topological space: A set equipped with a collection of subsets designated as open and satisfying three axioms. Connectedness is defined by how the open subsets of a topological space can partition it.
Connected component: A maximal connected subset of a topological space. Connected components partition any space into its largest connected pieces.
Real line: The set of real numbers with its usual topology and order. The real line is connected, and its intervals supply the basic model for the concept.
Path-connected space: A topological space in which every pair of points can be joined by a continuous path. Path-connectedness implies connectedness, but the topologist's sine curve shows the converse fails.
Borsuk–Ulam theorem: A theorem stating that every continuous map from an n-sphere to n-dimensional Euclidean space maps some antipodal pair to the same point. Its topological proof uses connectedness-related ideas to establish an unavoidable coincidence.
Clopen set: A subset that is both open and closed in a topological space. A space is connected precisely when its only clopen subsets are the empty set and the whole space.
Intermediate value theorem: A theorem stating that a continuous real-valued function on an interval takes every value between its endpoint values. It follows from the connectedness of intervals and characterizes a familiar consequence of connected domains.
Interval (mathematics): A subset of the real line containing every real number between any two of its points. Every interval in the real line is connected, including intervals with excluded endpoints.
Locally connected space: A topological space with a basis of connected neighborhoods at every point. Local connectedness concerns neighborhoods, whereas connectedness concerns the space as a whole.
Fixed-point theorem: A theorem guaranteeing a point left unchanged by a function under specified conditions. Connectedness is among the hypotheses that make several classical fixed-point results possible.

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