Connected space

Connected and disconnected subspaces of R²

From top to bottom: red space A, pink space B, yellow space C and orange space D are all connected spaces, whereas green space E (made of subsets E₁, E₂, E₃, and E₄) is disconnected. Furthermore, A and B are also simply connected (genus 0), while C and D are not: C has genus 1 and D has genus 4.

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 are used to distinguish topological spaces.

A subset of a topological space ${\displaystyle X}$ is a connected set if it is a connected space when viewed as a subspace of ${\displaystyle X}$.

Some related but stronger conditions are path connected, simply connected, and ${\displaystyle n}$-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.