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.

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