path

Let $I=[0,1] \subset \mathbb{R}$ and let $X$ be a topological space.

A continuous map $f:I\rightarrow X$ such that $f(0)=x$ and $f(1)=y$ is called a path in $X$ . The point $x$ is called the initial point of the path and $y$ is called its terminal point. If, in addition, the map is one-to-one, then it is known as an arc.

Sometimes, it is convenient to regard two paths or arcs as equivalent if they differ by a reparameterization. That is to say, we regard $f \colon I \to X$ and $g \colon I \to X$ as equivalent if there exists a homeomorphism $h \colon I \to I$ such that $h(0) = 0$ and $h(1) = 1$ and $f = g \circ h$ .

If the space $X$ has extra structure, one may choose to restrict the classes of paths and reparameterizations. For example, if $X$ has a differentiable structure, one may consider the class of differentiable paths. Likewise, one can speak of piecewise linear paths, rectifiable paths, and analytic paths in suitable contexts.

The space $X$ is said to be pathwise connected if, for every two points $x, y \in X$ , there exists a path having $x$ as initial point and $y$ as terminal point. Likewise, the space $X$ is said to be arcwise connected if, for every two distinct points $x, y \in X$ , there exists an arc having $x$ as initial point and $y$ as terminal point.

A pathwise connected space is always a connected space, but a connected space need not be path connected. An arcwise connected space is always a pathwise connected space, but a pathwise connected space need not be arcwise connected. As it turns out, for Hausdorff spaces these two notions coincide -- a Hausdorff space is pathwise connected iff it is arcwise connected.