homotópias

Homotopia is a concept in topology that describes a continuous deformation between two continuous maps from one topological space to another. Imagine you have two paths on a map. A homotopy between these paths would be a way to smoothly slide one path into the other without tearing it or jumping across any gaps. More formally, if you have two functions, f and g, both mapping from space X to space Y, a homotopy between f and g is a continuous function H from the product space X × [0, 1] to Y. The parameter 't' in H(x, t) represents the "time" of the deformation. At t=0, H(x, 0) = f(x) for all x in X, and at t=1, H(x, 1) = g(x) for all x in X. The intermediate values of t represent the stages of the continuous transformation. If such a continuous deformation exists, we say that f and g are homotopic. This concept is crucial for understanding the shape and properties of topological spaces, as it allows us to classify maps and spaces based on their deformation equivalence. For example, two loops on a sphere are homotopic if one can be continuously shrunk to a point while staying on the sphere. This is different from loops on a torus, where some loops cannot be shrunk to a point. Homotopy theory plays a significant role in various branches of mathematics, including algebraic topology and differential geometry.