= Connected component
{title2=$C_x$}
{wiki}
A <connected component> of a <topological space> $X$ is a nonempty <connected subset> that is maximal under <subset> inclusion. If $C$ is a <connected component> and $C\subseteq D\subseteq X$ with $D$ a <connected subset>, then $D=C$. Maximality means that no larger <connected subset> contains $C$; it does not mean largest <cardinality> among all <connected subsets>.
For example, the <connected components> of $[-2,2]\setminus\{0\}$ in the usual <subspace topology> of the <real line> are $[-2,0)$ and $(0,2]$. Each is a <real interval>, and no <connected subset of the real line> can meet both without containing the missing point $0$. For the <finite set> $\{-2,0,2\}$ in the same <subspace topology>, the three <connected components> are its <singleton sets>.
\Image[/connected-component-classification.png]
{title=An intact <real interval>, an <interval> with an interior point removed, and a <finite set> have one, two, and three <connected components> in the usual <subspace topology> of the <real line>}
{id=connected-component-classification}
Every point $x\in X$ lies in exactly one <connected component>, constructed by taking the <union>
$$
C_x=\bigcup\{A\subseteq X:x\in A\text{ and }A\text{ is connected}\}.
$$
All members contain $x$, so their <union> is <connected> by <connected union with a connected hub>. The <union> includes every <connected subset> through $x$, so it is maximal. Equivalently, define
$$
x\sim y\quad\Longleftrightarrow\quad
\text{some connected subset of }X\text{ contains both }x\text{ and }y.
$$
This is an <equivalence relation>: <singleton sets> give reflexivity, symmetry is immediate, and two <connected subsets> sharing the middle point have a <connected> <union>, giving transitivity. The <equivalence classes> are precisely the <connected components>. They form a <partition of a set>: they cover $X$ and any two are either equal or disjoint.<Point-Set Topology by Romyar Sharifi/Chapter 3: Connected and compact spaces>[Proposition 3.1.11, Definition 3.1.12 and Remark 3.1.16: maximal connected subsets and equivalence classes]{ref}
Every nonempty <connected subset> lies within one <connected component>. Moreover, every <connected component> is <closed> in $X$: <closure of a connected set> makes its <closure> <connected>, and maximality forces that <closure> to equal the original <connected component>. A <connected component> need not be <open>. For example, in $S=\{0\}\cup\{1/n:n=1,2,\ldots\}$ with its usual <subspace topology> from the <real line>, all <connected components> are <singleton sets>, but $\{0\}$ is not <open>: every <neighbourhood> of $0$ in $S$ contains some $1/n$.<Point-Set Topology by Romyar Sharifi/Chapter 3: Connected and compact spaces>[Lemma 3.1.13 and Example 3.1.14: closed components and a component which is not open]{ref}
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