homotópiatípusúak

Homotópiatípusúak are mathematical objects that are considered equivalent in a specific sense within homotopy theory. This equivalence is defined by the existence of a homotopy between the "maps" connecting them. In simpler terms, if you have two spaces, X and Y, they are homotopy equivalent if there are continuous functions f from X to Y and g from Y to X such that composing f with g (g after f) is continuously deformable to the identity function on X, and composing g with f (f after g) is continuously deformable to the identity function on Y.

This concept of homotopy equivalence is weaker than homeomorphism. Two spaces can be homotopy equivalent without being homeomorphic. For example, a single point and a circle are homotopy equivalent. This is because you can continuously map the point to any point on the circle, and then continuously map any point on the circle back to the original point in such a way that the compositions are homotopic to the identity. However, they are not homeomorphic because they have different topological properties (e.g., a circle is not contractible to a point).

Homotopy equivalence is a fundamental notion in algebraic topology as it allows for the classification of topological spaces up to this weaker form of equivalence. Many topological invariants, such as homotopy groups and homology groups, are preserved under homotopy equivalence, meaning that if two spaces are homotopy equivalent, they will have the same values for these invariants. This makes homotopy theory a powerful tool for studying the structure of topological spaces.