A continuum with no prime shape factors

As the main result of this paper, we prove that there exists a continuum with non-trivial shape without any prime factor. This answers a question of K. Borsuk [K. Borsuk, Concerning the notion of the shape of compacta, in: Proc. Internat. Symposium on Topology and Its Applications, Herceg-Novi, 1968, pp. 98–104]. We also show that for each integer n ⩾ 3 there exists a continuum X such that Sh ( X , x ) = Sh n ( X , x ) , but Sh ( X , x ) ≠ Sh n − 1 ( X , x ) . Therefore we obtain the negative answer to another question of K. Borsuk [K. Borsuk, Some remarks concerning the shape of pointed compacta, Fund. Math. 67 (1970) 221–240]: Does Sh ( X , x ) = Sh n ( X , x ) , for a compactum X and some integer n ⩾ 3 , implie that Sh ( X , x ) = Sh 2 ( X , x ) ?