Connected component ($C_x$)

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.

Every point $x\in X$ lies in exactly one connected component, constructed by taking the unionAll 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, defineThis 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.^[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 $S=\{0\}\cup \{1/n:n=1,2,\dots \}$ 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 $1/n$.^[ref]

Every connected component $C$ of a nonempty compact proper subset $D$ of a connected Hausdorff space meets its boundary. If not, the Šura-Bura lemma identifies $C$ as a quasi-component in the compact Hausdorff subspace $D$. Finitely many clopen neighborhoods separate it from the compact boundary, giving a nonempty relatively clopen $H\subseteq \mathrm{int}D$. Such $H$ is open in the ambient space, and is also closed there because $D$ is closed. It is proper because $D$ is proper. This contradicts connectedness.

The quasi-component of a point is the intersection of its clopen neighborhoods. Its connected component is always contained in it: a connected set meeting a clopen set cannot also meet its complement. Equality need not hold in noncompact Hausdorff spaces. The Šura-Bura lemma proves equality in compact Hausdorff spaces.

In a compact Hausdorff space, each connected component is its quasi-component and has a neighborhood basis of clopen sets. Let $Q$ be the intersection of clopen neighborhoods of $x$. It is compact. If it splits into two nonempty compact sets, normality separates them by disjoint open neighborhoods. The finite intersection characterization of compactness gives a finite intersection of clopen neighborhoods lying in their union. Its intersection with the neighborhood containing $x$ is then clopen and excludes part of $Q$, a contradiction. Thus $Q$ is connected and equals the component. Applying the same finite-intersection argument to any neighborhood of $Q$ proves the neighborhood-basis assertion. ^[ref].

Set $p=(0,0)$, $q=(1,0)$, ${\epsilon }_m=1/(m+2)$, and let $A_m$ be the polygonal arc joining $({\epsilon }_m,0)$, $({\epsilon }_m,m)$, $(1-{\epsilon }_m,m)$, and $(1-{\epsilon }_m,0)$. In the planar subspace $Z=\{p,q\}\cup {\bigcup }_m{A}_m$, each arc is clopen. The component of $p$ is therefore just $\{p\}$. Any clopen neighborhood of $p$ contains the left endpoints of all sufficiently high arcs, hence those entire connected arcs, and by closedness also the limit $q$ of their right endpoints. Thus its quasi-component is $\{p,q\}$. Higher arcs escape to infinity while their endpoints approach the two distinguished points.

The quasicomponent of a point is the intersection of all its clopen neighborhoods. Every connected set containing the point lies in its quasicomponent, since a clopen set cannot split a connected set. In a compact Hausdorff space, quasicomponents equal connected components by the Šura-Bura component lemma.

In a compact Hausdorff space, the quasicomponent $Q_x$ is connected and equals the connected component of $x$. If $Q_x=A\cup B$ were a nontrivial separation, disjoint open neighborhoods $U,V$ of its compact pieces leave a compact complement missing $Q_x$. Finitely many clopen neighborhoods of $x$ intersect to a clopen $N$ containing $Q_x$ inside $U\cup V$. The piece $N\cap U$ is then clopen in the whole space, contains $x$, and misses the nonempty other piece, contradicting the definition of $Q_x$. This also proves that any neighborhood of a component contains a clopen neighborhood of that component, by compactness of its complement.

The identity component is the connected component containing the identity element. It is a closed normal subgroup: products and inverses of connected sets through the identity remain in that component, and conjugation preserves it. It need not be open; in the additive group $\mathbb{Q}$ with its usual topology it is the nonopen singleton $\{0\}$.

The product of connected spaces is connected, so the product of the two connected components lies in the component of $(x,y)$. Conversely, continuous projections carry that component to connected sets in the factors, which must lie in $C_x$ and $C_y$. These two inclusions prove equality in arbitrary topological spaces.

An embedded copy of $S^{n-1}$ in ${\mathbb{R}}^n$ separates space into an inside and an outside. In particular, a connected closed surface embedded in ${\mathbb{R}}^3$ is two-sided and orientable.