Connected space

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.

Examples

 * 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.

Properties
The image of a connected space under a continuous map is again connected.

In conjunctions with the statement above, that the connected subsets of the real numbers with the Euclidean metric topology are the intervals, this gives the Intermediate Value Theorem.

Connected component
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.

Examples

 * A discrete space
 * The Cantor set
 * The rational numbers as a subspace of the real numbers with the Euclidean metric topology

Path-connected space
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 $$p: [0,1] \rightarrow X$$ such that p(0)=x and p(1)=y.

A path-connected space is connected, but not necessarily conversely.

Hyperconnected space
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.