Definition:Component (Topology)/Definition 1

Definition

Let $T = \struct {S, \tau}$ be a topological space.

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

$x \sim y$ if and only if $x$ and $y$ are connected in $T$.

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

From Connectedness of Points is Equivalence Relation, $\sim$ is an equivalence relation.

From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes.

These equivalence classes are called the (connected) components of $T$.

If $x \in S$, then the component of $T$ containing $x$ (that is, the set of points $y \in S$ with $x \sim y$) is denoted by $\map {\operatorname {Comp}_x} T$.

Also see

• Equivalence of Definitions of Component

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.5$: Components
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness