Path-connected space

A space is path connected when every two points can be joined by a continuous path within the space.

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 $\gamma :[0,1]\to X$.

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 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 is a maximal path-connected subset of a topological space.

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

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.