Vitaly V. Fedorchuk

Cellularity of covariant functors

Abstract We compute the cellularity of F ( X ) in terms of the cellularity of X for a class of covariant functors F including exp, λ and P . We also give a number of examples to test the sharpness of our results.

A differentiable manifold with noncoinciding dimensions

Abstract A differentiable n -manifold M n m , 4 ⩽ n m , with dimensions n = ind M n m m = dim M n m M n m = m + n − 2 is constructed under Jensens principle ♢. The space M n m is perfectly normal, countably compact and hereditarily separable.

On mappings of compact spaces into Cartesian spaces

Abstract Eilenberg proved that if a compact space X admits a zero-dimensional map f :X→Y , where Y is m -dimensional, then there exists a map h :X→I m+1 such that f×h :X→Y×I m+1 is an embedding. In this paper we prove generalizations of this result for σ -compact subsets of arbitrary spaces. An example of a compact space X and of a zero-dimensional σ -compact subset A⊂X is given such that for any continuous function f :X→ R which is one-to-one on the set A and any G δ -subset B of X with B⊃A the restriction f|B :B→ R has infinite fibers. This example is used to demonstrate that our results are sharp.

On a preservation of completeness of uniform spaces by the functor PscT

Abstract It is shown that the functor PscT of t -additive probability measures does not preserve completeness of uniform spaces. Namely, P T ( R c ) is not complete. On the other hand, assuming Martins Axiom, PscT(X) is complete for every complete uniform space X of uniform weight

On some dimensional properties of 4-manifolds

Abstract It is shown, under the assumption of Jensens principle ◊ , that if for a complex L with [L]≥[S 4 ] there exists a metrizable compactum whose extension dimension is [L] , then there exists a differentiable, countably compact, perfectly normal and hereditarily separable 4 -manifold whose extension dimension is also [L] .

The Suslin number of the functor of probability measures

Abstract We prove that c ( P ( X )) = c ( X ω ) for an arbitrary compact Hausdorff X .

Uniformly open mappings and uniform embeddings of function spaces

Abstract We study the class of uniformly open mappings between uniform spaces. Furthermore the Hausdorff uniformity on the space C b (X) of bounded continuous real-valued functions of a supercomplete uniform space X is investigated. With the help of our results we prove that for the Menger curve μ 1 the space C h b (μ 1 ) is a universal uniform coretract for the spaces C h b (X), where X is an arbitrary Peano continuum and C h b (X) denotes the completion of C B b (X).

Schauder's fixed point and amenability of a group

A criterion for existence of a fixed point for an affine action of a given group on a compact convex space is presented. From this we derive that a discrete countable group is amenable if and only if there exists an invariant probability measure for any action of the group on a Hilbert cube. Amenable properties of the group of all isometries of the Urysohn universal homogeneous metric space are also discussed.

On some categorical properties of uniform spaces of probability measures

Abstract We deal with the functor P β u : Unif → Unif of uniform spaces of probability measures, defined by Sadovnichy (1994). We show that there is a unique natural transformation T : S ∘ P gb u → P ∘ S , where S: Unif → c Unif is the functor of Samuel compactification. In our first main result (Theorem 4.3) it is established that for a uniform space (X, u ) the component T u of this natural transformation T is a homeomorphism iff u is a precompact uniformity. The second main result (Theorem 4.6) shows that there is no embedding U: Tych → Unif such that P β u ∘ U = U ∘ P β .

On spaces of σ-additive probability measures

Abstract The functor P σ of σ -additive probability measures on the category of Tychonoff spaces is investigated. It is shown that the space P σ ( X ) is Hewitt complete for every Tychonoff space X ; and P σ ( X ) is an AE-space of weight ⩽ ω 1 iff P σ ( X ) is an AE(0)-space of weight ⩽ ω 1 iff the Hewitt completion of X is an AE(0)-space of weight ⩽ ω 1 . It is shown that for every separable metrizable absolute Borel space X the space P σ ( X ω 1 ) is homeomorphic to P σ ( X ) ω 1 . In particular, P σ ( R ω 1 ) is homeomorphic to R ω 1 . We find conditions on a Tychonoff (uniform) space X under which P σ ( X ) is a Hewitt completion (respectively is naturally homeomorphic to the completion) of the (uniform) spaces P R ( X ) and P τ ( X ) of Radon and τ -additive probability measures on X .