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

Abstract

By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, $\displaystyle{\varprojlim (Y_i^{(n)},y_i^*)}$, where $(Y_i^{(n)},y_i^*)=\bigvee_{j\leq i}(X_j^{(n)},x_j^*)$ 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, $\pi_n^{top}(X,x^*)$, is a topological group for all $n\geq 2$ which is a partial answer to the open question whether $\pi_n^{top}(X,x^*)$ is a topological group for any space $X$ and $n\geq 1$. Moreover, we show that $\pi_n^{top}(X,x^*)$ is metrizable.