Homotopy of Paths
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 X is a family of paths ft : I → X indexed by I such that
- ft(0) = x0 and ft(1) = x1 are fixed.
- the map F : I × I → X given by F(s, t) = ft(s) is continuous.
The paths f0 and f1 connected by a homotopy are said to homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.
The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path f under this relation is called the homotopy class of f, often denoted .
Read more about this topic: Path (topology)
Famous quotes containing the word paths:
“Now it is the time of night
That the graves, all gaping wide,
Every one lets forth his sprite
In the church-way paths to glide.”
—William Shakespeare (1564–1616)