An infinite product of continua without nontrivial embeddings into itself

ABSTRACT. A continuum constructed by Cook is used to find a sequence (Xn) n of nondegenerate metrizable continua such that each continuous map (in particular: each embedding) of IlnX n into itself has the form Ilnf n, with each fn: Xn- • Xn the identity or constant. 1. Results. Ford and Heath ask in [2] if there is an infinite product of nondegenerate spaces which is not homeomorphic to any proper subspace. The following example is a strong affirmative answer. 1.1. EXAMPLE. There is a sequence (Xn) n of nondegenerate metrizable continua such that the identity is the only embedding of IInX n into itself. In fact, if f: IInX n- • IInX n is any continuous function, then f has the form IInf n, with each fn: Xn- • Xn either the identity or a constant. One ingredient of our example is a remarkable nondegenerate continuum M 1 constructed by Cook [1, Theorem 8], that has the property that for each subcontinuum K of M 1, every continuous map K- • M1 is either the identity or is constant. Since M 1 is nondegenerate i has a sequence (Xn) n of pairwise disjoint nondegenerate continua; clearly if n 4: m then every continuous map X n-•X m is constant; and every continuous map X n- • X n is the identity or is constant. That this is all we need to know about he Xn'S to establish Example 1.1 follows from our second ingredient. 1.2. THEOREM. Let II denote the product of an indexed family (X,y: 3 • C D of Hausdorff spaces. The following are equivalent: ( 1) if 3•,li • F are distinct, then every continuous map X3 •- • Xli is constant, ' an