In topology, a connected space is a topological space in which there is no (non-trivial) subset which is simultaneously open and closed. Equivalently, the only continuous function from the space to a discrete space is constant. A disconnected space is one which is not connected.
- The connected subsets of the real numbers with the Euclidean metric topology are the intervals.
- An indiscrete space is connected.
- A discrete space with more than one point is not connected.
The image of a connected space under a continuous map is again connected.
A connected component of a topological space is a maximal connected subset: that is, a subspace C such that C is connected but no superset of C is.
Totally disconnected space
A totally disconnected space is one in which the connected components are all singletons.
- A discrete space
- The Cantor set
- The rational numbers as a subspace of the real numbers with the Euclidean metric topology
A path-connected space is one in which for any two points x, y there exists a path from x to y, that is, a continuous function such that p(0)=x and p(1)=y.
A path-connected space is connected, but not necessarily conversely.
A hyperconnected space or irreducible space is one in which the intersection of any two non-empty open sets is again non-empty (equivalently the space is not the union of proper closed subsets).
A hyperconnected space is connected, but not necessarily conversely. Hyperconnectedness is open hereditary but not necessarily closed hereditary. Every topological space is homeomorphic to a closed subspace of a hyperconnected space.