Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

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).

• A separation of $X$ is an ordered pair $(U,V)$ of open, nonempty, disjoint subsets of $X$ with $U\cup V=X$.
• $X$ is disconnected when a separation of $X$ exists, and connected when none does.
• A subset $A\subseteq X$ is a connected subset of $X$ when the space $(A,{\mathcal{T}}_A)$ is connected, ${\mathcal{T}}_A$ being 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). "Disconnected subset" is read the same way.

Since $U$ and $V$ are complementary in $X$, each of them is closed as well as open; so a separation is the same thing as a partition of $X$ into two nonempty clopen pieces (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). The clopen subsets of $X$ are those that are both open and closed, and $\mathrm{\varnothing }$ and $X$ are always among them.

The empty space and the one-point space are connected in this library. Neither admits a separation: a separation requires two nonempty disjoint sets whose union is the whole space, and neither $\mathrm{\varnothing }$ nor a singleton can be written as such a union. So both are connected under the definition above, without any special clause. This is a live convention fork and the competing choice is recorded in Which conventions this page fixes: the empty space and the one-point space, separated sets against disjoint open sets, and what is not developed here; nothing on this page depends on which is taken except the reading of the word "connected" applied to those two spaces.

Connectedness is a property of a space, not of an ambient pair. The condition above mentions only $(X,\mathcal{T})$. When it is applied to $A\subseteq X$ it is applied to the space $(A,{\mathcal{T}}_A)$, so it does not change if $A$ is regarded as a subspace of some other space inducing the same topology on $A$; in particular a subset of $A$ is connected as a subset of $A$ exactly when it is connected as a subset of $X$, by transitivity of 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). This is why "connected" may be used of a subset with no ambient space named.

Spelled out for a subset. $A\subseteq X$ is disconnected exactly when there are open $U,V\subseteq X$ with

$A\subseteq U\cup V,U\cap A\mathrm{\ne }\mathrm{\varnothing },V\cap A\mathrm{\ne }\mathrm{\varnothing },U\cap V\cap A=\mathrm{\varnothing },$

because the open sets of $(A,{\mathcal{T}}_A)$ are precisely the traces $U\cap A$. Note the last condition: it asks $U$ and $V$ to be disjoint on $A$, not in $X$. Requiring $U\cap V=\mathrm{\varnothing }$ outright is a strictly stronger demand and is a different notion.

The two-point discrete space. Write $\mathbf{2}:=\{0,1\}$ with the discrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), in which every subset is open. A separation of $X$ is the same datum as a surjective continuous map $X\to \mathbf{2}$ (Continuity of a map of topological spaces at a point and globally): given $(U,V)$, the map sending $U$ to $0$ and $V$ to $1$ is continuous because the preimage of each of the four open subsets of $\mathbf{2}$ is one of $\mathrm{\varnothing }$, $U$, $V$, $X$; given a surjective continuous $\chi :X\to \mathbf{2}$, the pair $({\chi }^{-1}[\{0\}],{\chi }^{-1}[\{1\}])$ is a separation. This reformulation is proved as a theorem on this page and is recorded here only to name $\mathbf{2}$.

Separated sets. Two subsets $A_1,A_2\subseteq X$ are separated in $X$ when

$\overline{A_1}\cap A_2=\mathrm{\varnothing }\text{and}{A}_1\cap \overline{A_2}=\mathrm{\varnothing },$

closures taken in $X$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space, A point lies in the closure of $A$ iff every basic neighbourhood of it meets $A$; the closure is the smallest closed superset and equals $A$ together with its derived set). Separated sets are disjoint, since $A_1\subseteq \overline{A_1}$; the converse fails. This is verbatim the condition Separated sets, disconnection, and connected subset of $\mathbb{R}$ uses on the real line, transported to an arbitrary space, and the theorem relating it to the definition above is the next lemma on this page.

Totally disconnected spaces, and the empty case. The vocabulary for a space all of whose connected subsets are single points is fixed later on this page, together with the components; it is not defined here because it is stated in terms of components.

• Why "nonempty" and "disjoint" are both in the definition. Drop nonempty and every space with more than one open set is "disconnected" via $(X,\mathrm{\varnothing })$. Drop disjoint and $(X,X)$ separates every nonempty space. Drop open and every space with at least two points is separated by a point and its complement. Each of the four conditions is doing work, and the four together are the weakest demand under which the notion has the consequences proved on this page.

• Ordered pair, not unordered. A separation is written as a pair for convenience only; $(U,V)$ separates exactly when $(V,U)$ does, and no statement here distinguishes them.

• The relation to the real-line definition is proved, not assumed. Separated sets, disconnection, and connected subset of $\mathbb{R}$ defines connectedness of $E\subseteq \mathbb{R}$ by the absence of a partition into two nonempty separated sets, which is a condition on closures rather than on relatively open sets. That the two definitions agree is A subspace $A\subseteq X$ is disconnected exactly when $A=A_1\cup A_2$ with $A_1,A_2$ nonempty and separated in $X$, which is the criterion this library already uses on the real line together with The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$"; until those are proved, the two words are kept apart and no statement here quietly identifies them.