Definition:Component (Topology)/Definition 3

From ProofWiki
Jump to navigation Jump to search

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 $y$.


The component of $T$ containing $x$ is defined as:

the maximal connected set of $T$ that contains $x$.


Also see


Sources