Path-connected space

A topological space $X$ is path-connected if, for every $a,b\in X$, there is a continuous mapThis continuous path must stay within $X$ for its entire duration. Its parameter need not represent physical time, and continuity does not require a derivative or finite arc length. ^[ref]

In the closed interval $[0,1]$, any $a,b$ are joined by $\gamma (t)=(1-t)a+tb$. For $a=1/4$ and $b=3/4$, this becomes $\gamma (t)=1/4+t/2$, so $\gamma (0)=1/4$, $\gamma (1/2)=1/2$, and $\gamma (1)=3/4$. All its values stay in $[0,1]$. In the union $D=[-2,-1]\cup [1,2]$, however, a continuous path from $-1$ to $1$ would attain $0$ by the intermediate value theorem, although $0\notin D$. Thus $D$ is not path-connected.

Every path-connected space is a connected space. Proof outline: if $X=U\cup V$ admitted a separation of a topological space, choose $a\in U$ and $b\in V$. A continuous path between them would have nonempty open inverse images ${\gamma }^{-1}(U),{\gamma }^{-1}(V)$ with empty intersection and union $[0,1]$. This contradicts the connectedness of that closed interval.

The converse fails: the topologist's sine curve is connected but has two path components. Path connectedness is useful when we need an actual continuous path of allowed positions, rather than only the absence of a separation of a topological space. For instance, when modelling allowed positions as a topological space, membership in the same path component answers whether one can move continuously between two positions while remaining in that topological space.

The displayed subspace of the Euclidean plane is path-connected: each vertical rational tooth joins the horizontal line at height one. Delete $(r,1)$ for rational $r$. The residual tooth $V=\{r\}\times [0,1)$ cannot be joined by a path to the top line. Before a path first reaches height one, its first coordinate is a continuous rational-valued function on an interval and must therefore be constant; it would reach the deleted point.

The punctured space remains connected. Its portions $L,R$ on either side of $x=r$ are path-connected, and $V\subseteq \overline{L}\cap \overline{R}$ by the density of the rational numbers. Adjoining limit points to a connected set preserves connectedness: a separation would place the connected subset on one side and prevent the other side from containing its limit points. Hence $L\cup V$ and $R\cup V$ are connected and intersect in $V$. Their union is connected. This example distinguishes connectedness from path-connected using the loss of a single attachment point.

If $X$ had a separation into two disjoint nonempty open subsets, a continuous path joining points in the two subsets would pull it back to a separation of $[0,1]$. This contradicts connectedness of an interval. Thus every path-connected space is connected. The converse fails, for example for the topologist's sine curve with its limiting vertical segment.

If $f:X\to Y$ is continuous and $X$ is path-connected, then $f(X)$ is path-connected. Choose preimages of any two image points and a continuous path between them in $X$. Its composition with $f$ supplies the required path inside $f(X)$. The conclusion uses the subspace topology on the image, not an assumption that $f$ maps onto the whole codomain.

A path-connected region is multiply connected when some closed path cannot contract to a point within the region. An annulus and the complement of a solid torus are examples. The accessible region in the ideal magnetic Aharonov-Bohm effect has this obstruction. Such a region is not a simply connected space.

The map $F(s)=s{e}^{2\pi i/s}$ for $s>0$, $F(0)=0$, is continuous on $[0,1]$, since $|F(s)|=s$. Its image is the displayed spiral together with its endpoint. The image is compact and path-connected: between images of $s_0,s_1$ the explicit path is $F((1-u)s_0+u{s}_1)$.

The union of $\{0\}\times [0,1]$ with horizontal segments $[0,1]\times \{q\}$ for rational $q\in [0,1]$ is path-connected. Each point lies either on the vertical spine or on a segment meeting it. A path travels horizontally to the spine, vertically along it, and horizontally to its destination. This contrasts with disconnected rational-slope rays whose common origin has been excluded.

The complement of finitely many lines in ${\mathbb{R}}^3$ is path-connected, and any two points can be joined by two line segments. For endpoints $a,b$ outside the lines, exclude the finitely many affine planes through each endpoint and each removed line. Choose a bend point $c$ outside all those planes. If either segment met a removed line, the bend point would lie in its associated excluded plane, a contradiction.

For a countable set $T\subset {\mathbb{R}}^2$, join two points outside $T$ through an auxiliary point avoiding the countably many lines through either endpoint and a point of $T$. Such an auxiliary point exists: choose a line with direction outside the countable list of forbidden directions, then avoid its countably many intersections with those lines. The two segments avoid $T$, giving a continuous path.

A continuous path in $X$ is a continuous map $\gamma :[0,1]\to X$.

An open arc is a topological space homeomorphic to an open interval. Unlike an arc, which is homeomorphic to a closed interval, an open arc does not include its two endpoints.

The fixed-endpoint path space $\Omega (p,q)$ consists of paths in a manifold with specified initial and terminal points. Smooth, piecewise smooth and Sobolev $H^1$ versions have the same homotopy type in the usual manifold setting. Joining a path to a fixed path back from $q$ to $p$ gives a homotopy equivalence with the based loop space.

A continuous path with finite arc length on every compact parameter interval. Families of such paths are used in the measurable-density definition of extremal length.

A topological embedding $\gamma :S^1\to X$ of a circle into a topological space. On an oriented surface, an essential such curve is noncontractible; on a punctured surface, nonperipheral means that it does not surround only one puncture.

A simple closed curve in a connected oriented surface is nonseparating when cutting along it leaves a connected surface. A separating curve bounds a subsurface and represents zero in first homology. For a nonseparating curve, an arc between the two new boundary components closes to a curve meeting it once. Its algebraic intersection number of curves on an oriented surface is $\pm 1$, proving that the original curve represents a primitive homology class.

A collection of locally rectifiable paths, specified by their endpoints, winding or homotopy class when useful. Its extremal length compares its shortest metric length with total metric area.

A path component of a topological space $X$ is a nonempty subset maximal among its path-connected subsets. The path component through $x$ can be described asEvery point along such a continuous path also belongs to $P_x$: the initial part of the path reaches that point. Thus the paths witnessing membership stay inside $P_x$. This makes path components useful for deciding which allowed positions can be reached from one another by continuous motion.

Every path component lies inside one connected component, because every path-connected space is connected. Thus each connected component is a union of path components. In a locally path-connected space the two classifications agree, by connected locally path-connected spaces are path-connected. For a subset of the real line with its usual subspace topology, the classifications also agree: every connected component is a real interval, and interpolation supplies a continuous path within it.

The topologist's sine curve shows why the distinction matters in general. Use specifically the closed versionwith the subspace topology from the usual Euclidean plane. It has one connected component, $T$, but two path components, $G$ and $L$. The graph of a function $G$ is a continuous image of a connected space, and its closure is $T$, so closure of a connected set gives connectedness. Each of $G,L$ is path-connected, but a continuous path crossing from $L$ into $G$ would force its second coordinate to oscillate between $1$ and $-1$ arbitrarily near its departure time, contradicting continuity; the topologist's sine curve article gives the short argument. Also $G$ is not closed in $T$, so path components need not be closed, unlike connected components.^[ref]

Being joined by a continuous path is an equivalence relation. Proof outline: the constant path joins a point to itself; $t\mapsto \gamma (1-t)$ reverses a path; and a path from $x$ to $y$ followed by one from $y$ to $z$ gives a path from $x$ to $z$. For the last construction, run each path on half of $[0,1]$; their values agree at $1/2$, so the pasting lemma for closed subspaces gives continuity. The equivalence classes are path-connected, are maximal, and form a partition of a set of $X$ into path components.^[ref]

Every locally path-connected space is a locally connected space, but the converse can fail.

A space is locally path-connected when every point has a neighbourhood basis of path-connected open sets. Its path components are open.

In a locally path-connected space, every path component is open: a path-connected open neighbourhood of a point lies within that point's path component. The complement of a path component is a union of the other open path components, hence is also open. In a nonempty connected space, a nonempty subset that is both open and closed must be the whole space. Thus a connected space that is locally path-connected is path-connected.

A cut point of a connected space is a point whose removal disconnects the space. Every interior point of a closed interval is a cut point, whereas a two-dimensional closed disc has no cut point.