On Topological Homotopy Groups of $n$-Hawaiian like spaces

By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, ${\displaystyle \mathrm{}(Y_i^{(n)},y_i^{\ast })}$, where $(Y_i^{(n)},y_i^{\ast })={\bigvee }_{j\le i}(X_j^{(n)},x_j^{\ast })$ is the wedge of $X_j^{(n)}$'s in which $X_j^{(n)}$'s are $(n-1)$-connected, locally $(n-1)$-connected, $n$-semilocally simply connected and compact CW spaces. In this paper, first we show that the natural homomorphism $\displaystyle{\beta_n:\pi_n(X,)\rightarrow \varprojlim \pi_n(Y_i^{(n)},y_i^)}$ is bijection. Second, using this fact we prove that the topological $n$-homotopy group of an $n$-Hawaiian like space, ${\pi }_n^{top}(X,x^{\ast })$, is a topological group for all $n\ge 2$ which is a partial answer to the open question whether ${\pi }_n^{top}(X,x^{\ast })$ is a topological group for any space $X$ and $n\ge 1$. Moreover, we show that ${\pi }_n^{top}(X,x^{\ast })$ is metrizable.