A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if and then . Hence, being in the same component is an equivalence relation, and the equivalence classes are the connected components.

Using pathwise-connectedness, the pathwise-connected component containing is the set of all pathwise-connected to . That is, it is the set of such that there is a continuous path from to .

Technically speaking, in some topological spaces, pathwise-connected is not the same as connected. A subset of is connected if there is no way to write with and disjoint open sets. Every topological space decomposes into a disjoint union where the are connected. The are called the connected components of .

This entry contributed by Todd Rowland

CITE THIS AS:

Rowland, Todd. "Connected Component." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ConnectedComponent.html