A [topological space](/page/Topological%20Space) can fail to be connected in a way that no metric measurement detects at first glance. The set $[0,1] \cup [2,3] \subsetneq \mathbb{R}$ has no gap inside either interval, but a continuous path trying to move from the first interval to the second must jump across the missing open interval $(1,2)$. Connectedness is the language that detects whether a space has been split into two nonempty open pieces.

The basic question is not whether points are close together. It is whether the topology permits a global yes-or-no distinction that is locally invisible. If a space $X$ can be written as two disjoint nonempty open subsets, then a [continuous function](/page/Continuous%20Function) from $X$ to a two-point discrete space records which side a point lies on. Connected spaces are exactly the spaces where no such continuous binary invariant exists.

[example: Two Intervals Are Separated] Let $X = [0,1] \cup [2,3]$ with the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{R}$. Set $A=[0,1]$ and $B=[2,3]$. Both sets are nonempty because $0 \in A$ and $2 \in B$. They are disjoint: if $x \in A \cap B$, then $0 \le x \le 1$ and $2 \le x \le 3$, which would imply $2 \le x \le 1$, impossible. Also, \begin{align*} A \cup B &= [0,1] \cup [2,3] \\ &= X. \end{align*} To check openness in the subspace topology, compute the two intersections: \begin{align*} X \cap (-1,3/2) &= \bigl([0,1] \cup [2,3]\bigr) \cap (-1,3/2) \\ &= \bigl([0,1]\cap (-1,3/2)\bigr) \cup \bigl([2,3]\cap (-1,3/2)\bigr) \\ &= [0,1] \cup \varnothing \\ &= A, \end{align*} and \begin{align*} X \cap (3/2,4) &= \bigl([0,1] \cup [2,3]\bigr) \cap (3/2,4) \\ &= \bigl([0,1]\cap (3/2,4)\bigr) \cup \bigl([2,3]\cap (3/2,4)\bigr) \\ &= \varnothing \cup [2,3] \\ &= B. \end{align*} Since $(-1,3/2)$ and $(3/2,4)$ are open in $\mathbb{R}$, these equalities show that $A$ and $B$ are open in $X$. Thus $X$ is split into two nonempty disjoint open pieces, so the two intervals are separated even though each interval has no internal gap. [/example]

The example suggests the right obstruction. To define connectedness without mentioning distances, intervals, or paths, we first need a purely topological name for a split into two open sides. A separation is not just a partition into two sets; the two pieces must both be topologically visible from within the space.

[definition: Separation] Let $(X, \tau)$ be a topological space. A separation of $X$ is an ordered pair $(U,V)$ of subsets of $X$ such that: \begin{align*} U &\in \tau, \\ V &\in \tau, \\ U &\neq \varnothing, \\ V &\neq \varnothing, \\ U \cap V &= \varnothing, \\ U \cup V &= X. \end{align*} [/definition]

A separation is the precise failure mode. The primary definition names the spaces in which this failure mode is absent, so that later theorems can speak about continuity preserving a single unbroken whole.

## Definition

After naming separations, connectedness is the condition that rules them out. This is the useful direction: instead of trying to prove a space is made of one visible piece by inspecting all its points, we ask whether any proposed two-sided split can exist at all.

[definition: Connected Space] A topological space $(X, \tau)$ is connected if there is no separation of $X$. [/definition]

This definition is deliberately global. It does not ask whether nearby points can be joined, and it does not mention distances. It asks only whether the topology permits the whole space to be split into two simultaneously visible nonempty sides.

## Detecting Disconnection

Connectedness is often used through its failures. Once the primary definition is in place, it is useful to name the opposite condition and then translate separations into simpler tests involving one subset or one continuous binary-valued function.

### Disconnected Spaces

A space that admits a separation behaves like two topological worlds placed side by side. Naming that situation lets us discuss examples and counterexamples without carrying the two open pieces every time.

[definition: Disconnected Space] A topological space $(X, \tau)$ is disconnected if there exists a separation of $X$. [/definition]

The two sides of a separation determine each other by complementation. That observation turns disconnection into a question about a single subset whose presence and absence are both open conditions.

### Clopen Sets

A separation can be recorded by either one of its two sides, because the other side is its complement. This motivates a name for subsets whose membership and non-membership are both open conditions.

[definition: Clopen Subset] Let $(X, \tau)$ be a topological space. A subset $A \subset X$ is clopen in $X$ if $A$ is open in $X$ and $X \setminus A$ is open in $X$. [/definition]

A disconnected space may be hard to split if one searches directly for two open pieces. Clopen subsets isolate the obstruction in one place: a nonempty proper subset whose inside and outside are both topologically visible. The criterion below turns connectedness into the absence of exactly that obstruction.