Definition:Component (Topology)

Let $$T$$ be a topological space.

Let us define an equivalence relation $$\sim $$ on $$T$$ as follows:

$$x \sim y$$ iff $$x$$ and $$y$$ are both in the same connected subspace of $$T$$.

It is clear that $$\sim $$ is an equivalence: reflexivity and symmetry are obvious, and transitivity follows from Union of Connected Intersections is Connected.

The resulting equivalence classes are called the components of $$T$$.

Alternatively, a component of $$T$$ can be defined as a maximal connected subspace of $$T$$.

(Here, "maximal" is used in the sense that all connected subspaces of $$T$$ are themselves subsets of some component of $$T$$.)

The fact that these definitions are equivalent is demonstrated in Equivalence of Definitions of Component.