The properties and applications of relative homotopy

In the article we have introduced a notion of relative homotopy. Thus we have recalled a notion of relative retract and relative extension of continuous maps. We have studied some properties of a relative homotopy and we have proven that it is an equivalence relation. We have defined the notion of relative contractibility and we have proven the theorem on the extension of a relative homotopy. We have shown that a relative homotopy of continuous maps has similar properties to a homotopy of continuous maps. We have given a few applications of a relative homotopy that extend our knowledge of the fixed point theory, the theory of coincidence and the topological properties of acyclic sets.