Definition:Component (Topology)/Definition 1

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Definition

Let $T = \left({S, \tau}\right)$ 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 $\operatorname{Comp}_x \left({T}\right)$.


Also see


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