Homotopy properties of subsets of Euclidean spaces

This paper is devoted to prove several results concerning the homotopy groups of separable metric spaces that generalize some of the main results of [4] and [5] to homotopy groups. In particular, we focus on subspaces of Euclidean spaces. Among the results, we proposed a partial generalization of Shelah's Theorem to higher homotopy groups for noncompact spaces. Also, we discuss n-homotopically Hausdorff property, a separation axiom for n-loops introduced in [12], and conclude that each subset of Rn+1Rn+1 is n-homotopically Hausdorff. Moreover, the concept of a Hawaiian n  -wild point will be introduced that illustrates the complexity of homotopy group at that point. We show that any (n−1)(n−1)-connected locally (n−1)(n−1)-connected subspaces of Rn+1Rn+1 with uncountable nth homotopy group admit a Hawaiian n-wild point.Finally, we prove that n  th homotopy group of any (n−1)(n−1)-connected locally (n−1)(n−1)-connected subspace of Rn+1Rn+1 is free provided that it is n-semilocally simply connected, and then we study the free Abelian factor groups of the homotopy groups of these spaces.