The sequences of the sheaves of homotopy groups

Abstract

In this paper, we construct, the sequences of the sheaves of homotopy groups of a pair (XA) and of a triple (X,A,B), where X is a connected, locally path connected, and semilocally simply connected topological space and A, B are both open connected, locally path connected, and semilocally simply connected subspaces of X with $B subset A subset X$. Furthermore, we prove the existence of a homomorphism induced by a continuous map $f: (X,A)rightarrow (Y,B)$ (respectively, $f: (X,A,B)rightarrow (Y,C,D)$) between the sequences of the sheaves of homotopy groups of the pairs (X,A) and (Y,B) (respectively, of the triples (X,A,B) and (Y,C,D)).