The Julia set J x of the complex exponential function E x :z-* \e z for a real parameter A(0< A < 1/e) is known to be a Cantor bouquet of rays extending from the set A k of endpoints of / A to oo. Since A k contains all the repelling periodic points of £ A , it follows that J x = Cl (A A ). We show that A k is a totally disconnected subspace of the complex plane C, but if the point at oo is added, then A K = A k u {oo} is a connected subspace of the Riemann sphere C. As a corollary, A K has

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... logical dimension 1. Thus, oo is an explosion point in the topological sense for A K . It is remarkable that a space with an explosion point occurs 'naturally' in this way.