CZ EC H RE PU BL IC Means on scattered compacta

We prove that a separable Hausdorff topological space X containing a cocountable subset homeomorphic to [0, ω1] admits no separately continuous mean operation and no diagonally continuous n-mean for n ≥ 2. In this paper we construct a scattered compact space admitting no continuous mean operation, thus answering Problem 5 of [4]. By a mean operation on a set X we understand any binary operation μ : X × X → X such that μ(x, x) = x and μ(x, y) = μ(y, x) for all x, y ∈ X. If, in addition, the mean operation is associative, then it is called a semilattice operation. The mean operation is a partial case of an n-mean operation. A function μ : X → X defined on the nth power of a space X is called an n-mean operation (or briefly an n-mean) if (1) μ(x, . . . , x) = x for every x ∈ X and (2) μ is Sn-invariant in the sense that μ(xσ(1), . . . , xσ(n)) = μ(x1, . . . , xn) for any permutation σ of the set {1, . . . , n} and any vector (x1, . . . , xn) ∈ X. It is clear that a mean is the same as a 2-mean. The problem of detecting topological spaces with (or without) a continuous mean is classical in Algebraic Topology, see [1], [2], [3], [6], [7], [10]. It particular, due to Aumann [1], we know that for every n ≥ 1 the n-dimensional sphere admits no continuous mean. On the other hand, the 0-dimension sphere S = {−1, 1} trivially possesses such a mean. More generally, each zero-dimensional metrizable separable space, being homeomorphic to a subspace of the real line, admits a continuous semilattice operation. On the other hand, there are non-metrizable scattered compact Hausdorff spaces admitting no separately continuous semilattice operation. The simplest example is the compactification γN of the discrete space N of natural numbers whose remainder γN \ N is homeomorphic to the ordinal segment [0, ω1]. The existence of such a compactification γN follows from the famous Parovichenko theorem [9] (saying that any compact space of weight ≤ א1 is a continuous image of βN \ N). Another way to construct γN is as follows. Consider a family A = (Aα)α<ω1 of infinite subsets of N such that Aα ⊂∗ Aβ for any ordinals α < β. The almost inclusion Aα ⊂∗ Aβ means that Aα \ Aβ is finite. Now, consider the subalgebra B of P(N) generated by A ∪ { {n} } n∈N. Then γN is the space of ultrafilters on B. 1991 Mathematics Subject Classification. 54G12, 54H10, 22A26, 22A30, 54D30, 54D65.