On nondiscreteness of a higher topological homotopy group and its cardinality

Abstract

Here, we are going to extend Mycielski's conjecture to higher homotopy groups. Also, for an $(n-1)$-connected locally $(n-1)$-connected compact metric space $X$, we assert that $\pi^{top}_{n}(X)$ is discrete if and only if $\pi_{n}(X)$ is finitely generated. Moreover, $\pi^{top}_{n}(X)$ is not discrete if and only if it has the power of the continuum.