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.