Definition:Component (Topology)

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let the relation $\sim $ be defined on $T$ as follows:


 * $x \sim y$ $x$ and $y$ are connected in $T$.

That is, there exists a connected set of $T$ that contains both $x$ and $y$.

Also known as
A component of $T$ is also known as a connected component.

For simplicity of presentation, takes the position that a component is a connected set by definition, and so it is unnecessary and unwieldy to include the word connected when using it.

Also see

 * Equivalence of Definitions of Component
 * Definition:Path Component
 * Definition:Irreducible Component
 * Definition:Arc Component