Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Path (topology)
Summary
In mathematics, a path in a topological space is a continuous function from the closed unit interval into
Paths play an important role in the fields of topology and mathematical analysis.
For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space is often denoted
One can also define paths and loops in pointed spaces, which are important in homotopy theory. If is a topological space with basepoint then a path in is one whose initial point is . Likewise, a loop in is one that is based at .
A curve in a topological space is a continuous function from a non-empty and non-degenerate interval
A in is a curve whose domain is a compact non-degenerate interval (meaning are real numbers), where is called the of the path and is called its .
A is a path whose initial point is and whose terminal point is
Every non-degenerate compact interval is homeomorphic to which is why a is sometimes, especially in homotopy theory, defined to be a continuous function from the closed unit interval into
An or C0 in is a path in that is also a topological embedding.
Importantly, a path is not just a subset of that "looks like" a curve, it also includes a parameterization. For example, the maps and represent two different paths from 0 to 1 on the real line.
A loop in a space based at is a path from to A loop may be equally well regarded as a map with or as a continuous map from the unit circle to
This is because is the quotient space of when is identified with The set of all loops in forms a space called the loop space of
Homotopy
Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.
Specifically, a homotopy of paths, or path-homotopy, in is a family of paths indexed by such that
and are fixed.
the map given by is continuous.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.