homotópiai

Homotópia is a concept in topology that describes a continuous deformation between two continuous functions. More formally, given two continuous maps f and g from a topological space X to a topological space Y, a homotopy between f and g is a continuous map H: X × [0, 1] → Y such that H(x, 0) = f(x) for all x in X and H(x, 1) = g(x) for all x in X. The parameter t, which ranges from 0 to 1, can be thought of as time, and the map H(x, t) describes the continuous path traced by the point f(x) as it deforms into g(x).

Two maps are said to be homotopic if a homotopy exists between them. The relation of homotopy is an equivalence relation, meaning it is reflexive (any map is homotopic to itself), symmetric (if f is homotopic to g, then g is homotopic to f), and transitive (if f is homotopic to g and g is homotopic to h, then f is homotopic to h). This partitions the set of all continuous maps between two spaces into equivalence classes called homotopy classes.

Homotopy is a fundamental tool in algebraic topology. It allows mathematicians to classify topological spaces and understand their global properties. For instance, if two spaces have a different number of "holes," they are generally not homotopic. The concept of homotopy equivalence is a weaker form of homeomorphism, stating that two spaces are essentially the same up to continuous deformation. This notion is crucial for understanding the shape of objects in a way that is invariant under stretching and bending.