homotopit

A homotopie is a continuous deformation between two continuous maps. More formally, if $f$ and $g$ are two continuous maps from a topological space $X$ to a topological space $Y$, a homotopie between $f$ and $g$ is a continuous map $H: X \times [0, 1] \to Y$ such that for all $x \in X$, $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$. The parameter $t \in [0, 1]$ can be thought of as "time," where $H(x, t)$ represents the position of the point $x$ at time $t$ during the deformation from $f$ to $g$.

The concept of homotopie is fundamental in algebraic topology. Two maps are said to be homotopic if there exists a homotopie between them. The relation of being homotopic is an equivalence relation. This means that any two maps are either homotopic or not homotopic, and the relation is reflexive, symmetric, and transitive.

Homotopie is particularly useful for classifying topological spaces. For example, if two spaces are homeomorphic, they are also homotopic. However, the converse is not always true; two spaces can be homotopic without being homeomorphic. Spaces that are equivalent up to homotopie are considered to be the same from an algebraic topological perspective.

A key application of homotopie is in the study of fundamental groups. The fundamental group $\pi_1(X, x_0)$ of a topological space $X$ with a base point $x_0$ consists of homotopy classes of closed loops based at $x_0$. Two loops are in the same homotopy class if one can be continuously deformed into the other while keeping the base point fixed. This idea of deformation is precisely what homotopie captures.