A [connected](/page/Connectedness) space cannot be split into two disjoint open pieces. But this algebraic condition — the absence of a separation — does not guarantee that you can "walk" continuously from one point to another. The topologist's sine curve is connected (it cannot be partitioned into two open halves), yet no continuous path joins the oscillatory graph to the vertical segment at the origin. The fundamental difficulty is that connectedness is a *global* condition on the open-set structure, while the ability to travel between points is a *local-to-global* geometric property that requires building continuous curves.

**Path connectedness** is the stronger, more geometric notion: a space is path-connected if every pair of points can be joined by a continuous map from the unit interval. This condition is strictly stronger than connectedness — every path-connected space is connected, but not conversely — and it is the notion that underpins most geometric and algebraic-topological constructions. The [fundamental group](/page/Fundamental%20Group), covering space theory, and the classification of surfaces all begin with paths. Homotopy theory is, at its foundation, the study of when two paths can be continuously deformed into one another.

The gap between the two notions is closed in many natural settings. For [open subsets of Euclidean space](/page/Connectedness), connectedness and path connectedness are equivalent — a fact that depends critically on the local convexity of $\mathbb{R}^n$. More generally, the two notions coincide for any space that is locally path-connected, a condition that captures precisely the local structure needed to piece together short paths into global ones.

[example: The Topologist's Sine Curve Is Not Path-Connected] The canonical example demonstrating the gap between connectedness and path connectedness is the **topologist's sine curve**. Define $S \subset \mathbb{R}^2$ 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 graph $\Gamma := \{(x, \sin(1/x)) : x \in (0, 1]\}$ is the continuous image of the connected interval $(0, 1]$, hence connected. Every point of the vertical segment $I := \{0\} \times [-1, 1]$ is a limit of points of $\Gamma$: for any $y \in [-1, 1]$, the function $\sin$ takes the value $y$ infinitely often on $(0, \varepsilon)$ for every $\varepsilon > 0$, so there exist $x_k \to 0^+$ with $\sin(1/x_k) = y$. Therefore $S = \overline{\Gamma}$, and since the [closure of a connected set is connected](/page/Connectedness), $S$ is connected. However, $S$ is **not** path-connected. Suppose for contradiction that $\gamma: [0, 1] \to S$ is a continuous path with $\gamma(0) = (0, 0) \in I$ and $\gamma(1) = (1, \sin 1) \in \Gamma$. Write $\gamma(t) = (\gamma_1(t), \gamma_2(t))$. Since $\gamma_1$ is continuous and $\gamma_1(0) = 0$, the set $\{t \in [0, 1] : \gamma_1(t) = 0\}$ is closed; let $t_0$ be its supremum. Then $\gamma_1(t_0) = 0$, and for $t > t_0$ we have $\gamma_1(t) > 0$, so $\gamma_2(t) = \sin(1/\gamma_1(t))$. As $t \to t_0^+$, we have $\gamma_1(t) \to 0^+$, so $\sin(1/\gamma_1(t))$ oscillates between $-1$ and $1$ without converging. But continuity of $\gamma_2$ requires $\gamma_2(t) \to \gamma_2(t_0)$, a contradiction. The essential mechanism is that the graph oscillates with unbounded frequency near $x = 0$: any path approaching the vertical segment from the graph side must traverse infinitely many oscillations in finite "time," which is incompatible with continuity. This phenomenon cannot occur in an open subset of $\mathbb{R}^n$, where every point has a convex (hence path-connected) neighbourhood. [/example]

## Definition

The definition of path connectedness requires two preliminary notions: what a path is, and what it means to concatenate paths.

[definition: Path] Let $X$ be a [topological space](/page/Topology). A **path** in $X$ is a [continuous](/page/Continuity) map $\gamma: [0, 1] \to X$. The point $\gamma(0)$ is the **initial point** (or **start point**) and $\gamma(1)$ is the **terminal point** (or **end point**). We say $\gamma$ is a path **from** $\gamma(0)$ **to** $\gamma(1)$. For points $x, y \in X$, a path from $x$ to $y$ is a continuous map $\gamma: [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$. [/definition]

The choice of $[0, 1]$ as the domain is a convention. Any closed interval $[a, b]$ with $a < b$ would serve equally well, since $[a, b]$ is homeomorphic to $[0, 1]$ via the affine map $t \mapsto (t - a)/(b - a)$. The unit interval is chosen for definiteness.

[remark: Paths vs. Curves] A path is a *map*, not a *set*. The **image** $\gamma([0, 1]) \subset X$ is sometimes called the **trace** or **arc** of the path, but this is a subset of $X$, not a path itself. Two paths can have the same image but differ as maps — for instance, a path traversed at constant speed and the same path traversed with a pause halfway through are different paths (as functions) with identical images. This distinction becomes critical in the [fundamental group](/page/Fundamental%20Group), where paths are considered up to homotopy, not just up to their images. [/remark]

Paths can be composed end-to-end, reversed, and degenerated to a point. These operations give the collection of paths an algebraic structure that is not quite a group (concatenation is not associative on the nose, and reversal is not a strict inverse), but becomes one after passing to homotopy classes.

[definition: Path Operations] Let $X$ be a topological space. **Concatenation.** If $\gamma: [0, 1] \to X$ is a path from $x$ to $y$ and $\delta: [0, 1] \to X$ is a path from $y$ to $z$, their **concatenation** $\gamma * \delta: [0, 1] \to X$ is defined by: \begin{align*} (\gamma * \delta)(t) := \begin{cases} \gamma(2t) & \text{if } 0 \le t \le \tfrac{1}{2}, \\ \delta(2t - 1) & \text{if } \tfrac{1}{2} \le t \le 1. \end{cases} \end{align*} This is continuous by the [pasting lemma](/page/Continuity): the two pieces agree at $t = 1/2$ (where $\gamma(1) = y = \delta(0)$), and each piece is continuous on a closed subset of $[0, 1]$. **Reverse.** If $\gamma: [0, 1] \to X$ is a path from $x$ to $y$, the **reverse path** $\bar{\gamma}: [0, 1] \to X$ is defined by $\bar{\gamma}(t) := \gamma(1 - t)$. This is a path from $y$ to $x$. **Constant path.** For any $x \in X$, the **constant path at $x$** is $c_x: [0, 1] \to X$ defined by $c_x(t) := x$ for all $t$. [/definition]

With these ingredients, we can state the central definition.

[definition: Path-Connected Space] A topological space $X$ is **path-connected** if for every pair of points $x, y \in X$, there exists a path from $x$ to $y$: a continuous map $\gamma: [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$. A subset $A \subset X$ is **path-connected** if $A$ is path-connected in the [subspace topology](/page/Topology) inherited from $X$. [/definition]

[remark: The Relation "Connected by a Path" Is an Equivalence Relation] For a topological space $X$, define the relation $x \sim y$ if there exists a path in $X$ from $x$ to $y$. This is an equivalence relation: - **Reflexivity:** the constant path $c_x$ is a path from $x$ to $x$. - **Symmetry:** if $\gamma$ is a path from $x$ to $y$, then $\bar{\gamma}$ is a path from $y$ to $x$. - **Transitivity:** if $\gamma$ is a path from $x$ to $y$ and $\delta$ is a path from $y$ to $z$, then $\gamma * \delta$ is a path from $x$ to $z$. The equivalence classes of this relation are the **path components** of $X$, and $X$ is path-connected if and only if it has exactly one path component. [/remark]

## Path Components

The equivalence relation "connected by a path" partitions any topological space into maximal path-connected subsets. Understanding the structure of these pieces — and how they compare to the [connected components](/page/Connectedness) of the same space — clarifies the precise relationship between the two notions of "one piece."

[definition: Path Component] Let $X$ be a topological space and $x \in X$. The **path component** of $x$ is the set: \begin{align*} P_x := \{y \in X : \text{there exists a path in } X \text{ from } x \text{ to } y\}. \end{align*} Equivalently, $P_x$ is the equivalence class of $x$ under the relation "connected by a path." [/definition]

Each path component is path-connected (by transitivity of the path relation) and is the largest path-connected subset containing $x$: if $A \subset X$ is path-connected and $x \in A$, then $A \subset P_x$.

The fundamental comparison between path components and connected components is as follows. Since path connectedness implies connectedness, every path component is contained in a connected component. But a connected component may contain several path components.

[quotetheorem:1055]