Homotopy

A homotopy is a continuous deformation from one continuous map to another, represented by a map parametrized by an interval. It expresses when two maps are continuously deformable into each other.

Topological space: A set equipped with a collection of open subsets that specifies which subsets count as neighborhoods. The domain and codomain of a homotopy are topological spaces.
Homotopy of paths: A continuous deformation between paths, often required to keep their endpoints fixed. Path homotopies are the basic case used to define fundamental groups.
Fundamental group: The group of based loops in a space, taken up to homotopy, with concatenation as its operation. It records which loops can or cannot be deformed into one another.
Homeomorphism: A continuous bijection between topological spaces whose inverse is also continuous. Homeomorphism requires exact topological equivalence, stronger than homotopy equivalence.
Continuous function: A function between topological spaces whose inverse images of open sets are open. Each fixed parameter value in a homotopy gives a continuous function.
Homotopy relative to a subspace: A homotopy that leaves every point of a specified subspace fixed throughout the deformation. This condition formalizes deformations that preserve selected points or boundaries.
Homotopy groups: Groups that classify homotopy classes of maps from spheres into a pointed space. They extend the loop-classification idea to maps from higher-dimensional spheres.
Isotopy: A continuous family of embeddings connecting two embeddings while every intermediate map remains an embedding. An isotopy prevents intermediate maps from collapsing points, unlike a general homotopy.
Unit interval: The real interval from zero to one, commonly used to parametrize continuous processes. Its parameter selects the map at each stage of the deformation.
Path-connected space: A topological space in which every pair of points can be joined by a continuous path. Paths are maps from an interval, and their homotopies compare such maps.

Linked from 11 pages