Roman Pol

A weakly infinite-dimensional compactum which is not countable-dimensional

A compact metric space is constructed which is neither a countable union of zero-dimensional sets nor has an essential map onto the Hilbert cube. We consider only separable metrizable spaces and a compactum means a compact space. A space is countable-dimensional if X = U7 1 Xi with Xi zero-dimensional; a space X is weakly infinite-dimensional if for each countable family {(Ai, B,): i = 0, 1, . .} of pairs of disjoint closed sets in X there are partitions Si between A, and Bi (i.e., closed sets separating Ai and Bi in X) with f70 Si = 0 [A-P, Chapter 10, ?47], [N]. Countable-dimensional spaces are weakly infinite-dimensional2 and an old open question of P. S. Aleksandrov [Al, ?4, Hypothesis] (cf. also [A2], [A-P, Chapter 10], [S], [N, Problem 13-7]) asked whether the converse is true for compacta.3 In this note we present a counterexample, i.e., we describe a compactum X with the properties indicated in the title. The existence of such an X is an easy consequence of the following lemma. LEMMA. There exists a topologically complete space Y which is totally disconnected but not countable-dimensional (not even weakly infinite-dimensional). The existence of such a space Y follows immediately from a construction in [R-S-W] (see also Comment A). More specifically, if one performs the construction in Example 4.5 of [R-S-W] using, as indicated in Remark 4.4, the Hilbert cube instead of the n + I-dimensional cube, then one obtains a compactum M and a continuous map p: M -> A onto the Cantor set A such that each subset of M which maps onto A is not weakly infinite-dimensional (see Proposition 3.4 and Remark 4.1 of [R-S-W]). It is, however, well known that in this situation there exists a G6-set Y c M which intersects each fiber p l(t) in exactly one point [B, p. 144, Exercise 9a], [Ku2, Chapters IV, IX], and this is the space Y we need. Received by the editors April 21, 1980. 1980 Mathematics Subject Classification Primary 54F45. IThis paper was written while the author was visiting the University of Washington. 2This follows from the fact [H-W, Chapter II, ?2, F] that, given two closed disjoint sets A, B c X and a zero-dimensional set E c X, there is a partition in X between A and B disjoint from E; cf. also [H-W, Chapter IV, ?6, Al. 3For nonmetrizable compact spaces, a counterexample was recently constructed by Fedorcuk [F]. i 1981 American Mathematical Society 0002-9939/81 /0000-0374/

Note on function spaces with the topology of pointwise convergence

01.75

THE BAIRE CATEGORY THEOREM IN PRODUCTS OF LINEAR SPACES AND TOPOLOGICAL GROUPS

Abstract. We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class

A remark on A-weakly infinite-dimensional spaces☆

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Note on pointwise convergence of sequences of analytic sets

of

A metric space with the Haver property whose square fails this property

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A hereditarily normal strongly zero-dimensional space with a subspace of positive dimension and an N -compact space of positive dimension

-functions. The proof is based on a weak zero-density estimate near the 1-line and on a simple almost periodicity argument. We also give a conditional proof of the non-vanishing on the 1-line for every

On some problems concerning Borel structures in function spaces

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Weak infinite-dimensionality in Cartesian products with the Menger Property

-function in

A complete C-space whose square is strongly infinite-dimensional

\mathcal{S}