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+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)-connected locally (n−1)-connected subspaces of Rn+1 with uncountable nth homotopy group admit a Hawaiian n-wild point. Finally, we prove that n th homotopy group of any (n−1)-connected locally (n−1)-connected subspace of Rn+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.

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