One of the most basic geometric intuitions is that a space can consist of "one piece" or "several pieces." A circle is one piece; a pair of disjoint circles is two pieces. The challenge for topology is to capture this distinction using only the language of open sets — without reference to distance, coordinates, or any ambient space.

The difficulty is more subtle than it first appears. In a [metric space](/page/Metric%20Space), one might try to define "one piece" by saying that any two points can be joined by a curve. This works well for open subsets of $\mathbb{R}^n$, but it fails for general topological spaces. The **topologist's sine curve** — the closure of the graph of $\sin(1/x)$ — is a subset of $\mathbb{R}^2$ that is intuitively "one piece" (it cannot be split into two open halves), yet no continuous path joins the oscillatory part to the vertical segment at the origin. This gap between "cannot be separated" and "can be joined by a path" is the central tension of the theory.

[example: The Topologist's Sine Curve] Define the set $S \subset \mathbb{R}^2$ by: \begin{align*} S := \left\{\left(x, \sin\frac{1}{x}\right) : x \in (0, 1]\right\} \cup \left(\{0\} \times [-1, 1]\right). \end{align*} The first component is the graph of the function $g: (0, 1] \to \mathbb{R}$, $g(x) = \sin(1/x)$, which oscillates with increasing frequency as $x \to 0^+$. The second component is the vertical segment $I = \{0\} \times [-1, 1]$. Every point of $I$ is a limit point of the graph: for any $(0, y)$ with $y \in [-1, 1]$, there exists a sequence $x_k \to 0^+$ with $\sin(1/x_k) = y$ (since $\sin$ takes every value in $[-1, 1]$ infinitely often on any interval $(0, \varepsilon)$). Therefore $S = \overline{\{(x, \sin(1/x)) : x \in (0, 1]\}}$, and $S$ is the closure of a connected set (the continuous image of the connected interval $(0, 1]$). However, $S$ is **not** path-connected. Suppose for contradiction that there exists a [continuous](/page/Continuity) path $\gamma: [0, 1] \to S$ with $\gamma(0) = (0, 0)$ and $\gamma(1) = (1, \sin 1)$. Write $\gamma(t) = (\gamma_1(t), \gamma_2(t))$. Let $t_0 = \sup\{t \in [0, 1] : \gamma_1(t) = 0\}$. By continuity of $\gamma_1$ and the fact that $\{t : \gamma_1(t) = 0\}$ is closed, $\gamma_1(t_0) = 0$. For $t > t_0$, we have $\gamma_1(t) > 0$, so $\gamma_2(t) = \sin(1/\gamma_1(t))$. As $t \to t_0^+$, the values $\gamma_1(t) \to 0^+$, so $\sin(1/\gamma_1(t))$ oscillates between $-1$ and $1$ without settling. But $\gamma_2$ is continuous, so $\gamma_2(t) \to \gamma_2(t_0)$ must exist — a contradiction. This space therefore demonstrates that the two natural formalisations of "one piece" — connectedness and path-connectedness — genuinely differ. [/example]

The theory of connectedness addresses this by introducing two separate notions. **Connectedness** (the algebraic-topological notion) captures the idea that a space cannot be separated into two open halves. **Path-connectedness** (the geometric notion) captures the idea that any two points can be joined by a continuous curve. Path-connectedness implies connectedness, but not conversely — and understanding when the two notions coincide is one of the main questions in the subject.

## Definition

The fundamental definition avoids any mention of paths or distances. Instead, it characterises "one piece" by the absence of a certain kind of partition.

[definition: Connected Topological Space] A [topological space](/page/Topology) $(X, \tau)$ is **connected** if there do not exist non-empty [open sets](/page/Open%20Set) $U, V \in \tau$ with: \begin{align*} U \cap V = \varnothing \quad \text{and} \quad U \cup V = X. \end{align*} Such a pair $(U, V)$ is called a **separation** of $X$. Equivalently, $X$ is connected if and only if the only subsets of $X$ that are both open and [closed](/page/Closed%20Set) (i.e., **clopen**) are $\varnothing$ and $X$ itself. A subset $A \subset X$ is **connected** if $A$ is connected in the subspace topology inherited from $X$. [/definition]

The equivalence of the two formulations is immediate: if $(U, V)$ is a separation, then $U$ is clopen (it is open by assumption and closed because its complement $V$ is open). Conversely, if $U$ is a non-trivial clopen set ($U \neq \varnothing$, $U \neq X$), then $(U, X \setminus U)$ is a separation.

[remark: The Empty Space] The empty space $\varnothing$ satisfies the definition vacuously: there is no separation because there are no non-empty open sets at all. By convention, $\varnothing$ is considered connected in most references, though some authors exclude it. We follow the convention that $\varnothing$ is connected. [/remark]

The definition is purely negative — it says what a connected space is *not* (separable into two open halves), rather than what it *is*. This makes connectedness difficult to verify directly. The following equivalent characterisations provide more workable criteria.

[quotetheorem:294]

Characterisation (3) is often the most useful in practice. To show that a space is connected, it suffices to show that every continuous integer-valued function on it is constant. To show that a space is *disconnected*, it suffices to exhibit a continuous surjection onto the two-point discrete space $\{0, 1\}$.

Characterisation (2) is the topological generalisation of the [Intermediate Value Theorem](/page/Continuity): a continuous real-valued function on a connected space cannot "jump" from one value to another without passing through every intermediate value. In fact, for subsets of $\mathbb{R}$, the Intermediate Value Theorem is precisely the statement that connected subsets are intervals — a result we establish below.

## Connected Subsets of the Real Line

The most concrete setting for connectedness is the real line $\mathbb{R}$ with its standard [topology](/page/Topology). Here, the topological notion of "one piece" coincides exactly with the order-theoretic notion of convexity: a subset of $\mathbb{R}$ is connected if and only if it is an interval.

This result is fundamental because it provides the bridge between the abstract topological definition and the classical Intermediate Value Theorem from real analysis. It also serves as the base case for many inductive arguments about connectedness in $\mathbb{R}^n$.

[quotetheorem:295]

The "if" direction (intervals are connected) is the essence of the Intermediate Value Theorem. The "only if" direction (connected subsets are intervals) uses the completeness of $\mathbb{R}$: if $X$ contains points $a < b$ but misses some $c \in (a, b)$, then $X \cap (-\infty, c)$ and $X \cap (c, \infty)$ form a separation of $X$.

This characterisation fails in the rationals $\mathbb{Q}$. The set $\mathbb{Q}$ is totally disconnected: the only connected subsets are singletons and the empty set. Indeed, for any two rationals $a < b$, the irrational number $\sqrt{2}$ (or any irrational between $a$ and $b$) provides a point $c \notin \mathbb{Q}$ that splits $\{q \in \mathbb{Q} : a \le q \le b\}$ into two relatively open halves $\{q : a \le q < c\}$ and $\{q : c < q \le b\}$. The completeness of $\mathbb{R}$ — the fact that every bounded set has a supremum — is what prevents this splitting.

[example: A Connected Subset of $\mathbb{R}$ That Is Not Path-Connected Does Not Exist] In $\mathbb{R}$, connectedness and path-connectedness coincide for all subsets. If $X \subset \mathbb{R}$ is connected, then $X$ is an interval, and any two points $a, b \in X$ are joined by the path $\gamma: [0, 1] \to X$ defined by $\gamma(t) = (1-t)a + tb$. The image of $\gamma$ is the interval $[\min(a,b), \max(a,b)] \subset X$ (since $X$ is convex). This is a special feature of $\mathbb{R}$. In $\mathbb{R}^2$, the topologist's sine curve (discussed above) shows that the two notions genuinely differ. [/example]