Connected components, quasicomponents, and totally disconnected spaces

Let $(X, \mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), with subsets carrying the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace) and connectedness as in Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets. Let $x \in X$.

• The connected component of $x$ is $C(x) \;:=\; \bigcup \{\, A \subseteq X : x \in A \text{ and } A \text{ is connected} \,\} .$ A component of $X$ is a set of the form $C(x)$ for some $x \in X$.
• The quasicomponent of $x$ is $Q(x) \;:=\; \bigcap \{\, K \subseteq X : x \in K \text{ and } K \text{ is clopen in } X \,\} .$ A quasicomponent of $X$ is a set of the form $Q(x)$.
• $X$ is totally disconnected when $C(x) = \{x\}$ for every $x \in X$.

Both are well posed, and the obligations are discharged here. The family united in the definition of $C(x)$ is nonempty, since the singleton $\{x\}$ is connected: a singleton admits no separation, a separation requiring two disjoint nonempty pieces. Every member of that family contains $x$, so A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member claim 1 applies and $C(x)$ is connected; being a union of every connected set through $x$, it contains each of them, so $C(x)$ is the largest connected subset of $X$ containing $x$. The family intersected in the definition of $Q(x)$ is nonempty as well, since $X$ itself is clopen (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), so the intersection is a set; it contains $x$, every member doing so.

Both notions are defined by a property of $X$, not of an ambient space. A component of a subspace $S \subseteq X$ means a component of the space $S$, and is written $C_S(y)$ when the space needs naming. The same holds for quasicomponents.

Totally disconnected, spelled out. $X$ is totally disconnected exactly when every connected subset of $X$ has at most one point: if some connected $A$ had two points $x \ne y$ then $A \subseteq C(x)$ would give $C(x) \ne \{x\}$, and conversely if $C(x) \ne \{x\}$ then $C(x)$ is a connected set with at least two points. The empty space is totally disconnected, having no point to test.

A discrete space is totally disconnected. Let $X$ carry the discrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) and let $A \subseteq X$ have two distinct points $x, y$. Every subset of $A$ is open in $A$, so $(\{x\}, A \setminus \{x\})$ is a pair of open, disjoint, nonempty sets covering $A$, that is a separation. Hence no connected subset has two points and every component is a singleton. The converse fails: total disconnectedness does not force the topology to be discrete.

• Why two notions and not one. The component of $x$ is built from the connected sets through $x$ and the quasicomponent from the clopen sets containing $x$. One is an inner approximation, assembled from below out of pieces known to be connected; the other is an outer approximation, cut down from above by every partition of $X$ into two clopen pieces. They always satisfy $C(x) \subseteq Q(x)$, and they can differ; both facts are theorems on this page, and the difference is exactly the gap between "cannot be split by a clopen set" and "is connected".

• Quasicomponents are what a separation argument actually produces. A proof that two points cannot be separated typically produces a clopen set containing both or neither, which is a statement about $Q$, not about $C$. Naming the weaker notion keeps such an argument honest instead of letting it be read as a connectedness claim.

• The definition of totally disconnected is stated with components, not with quasicomponents. The condition "every quasicomponent is a singleton" is a different and strictly stronger property, usually called total separatedness. Nothing on this page asserts that the two agree.