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 nor connected.

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.