Elżbieta Pol

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

Haver introduced the following property of metric spaces (X,d): for each sequence ∈ 1 ,∈ 2 ;.. of positive numbers there exist collections ν 1 , ν 2 ,. of open subsets of X, the union ∪ i ν i of which covers X, such that the members of ν i are pairwise disjoint and every member of ν i has diameter less than ∈ i . We construct two separable complete metric spaces (X 0 , d 0 ), (X 1 ,d 1 ) with the Haver property such that d 0 , d 1 generate the same topology on X 0 n X 1 ≠ O, O, but (X 0 ∩ X 1 ,max(d 0 , d 1 )) fails this property. In particular, the square of a separable complete metric space with the Haver property may fail this property. Our results answer some questions posed by Babinkostova in 2007.

Spaces whose n th power is weakly infinite-dimensional but whose (n+1) th power is not

For every natural number n we construct a metrizable separable space Y such that yn is weakly infinite-dimensional (moreover, is a C-space) but yn+1 is strongly infinite-dimensional.

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

In this paper we give a solution of an old Cech’s problem on dimension by constructing a hereditarily normal strongly zero-dimensional space containing a subspace of positive dimension. We give also an example of an N-compact space of positive dimension.

On infinite-dimensional Cantor manifolds

Abstract In this paper we construct a weakly infinite-dimensional compactum which cannot be separated by any hereditarily weakly infinite-dimensional compactum and a family of uncountably many hereditarily infinite-dimensional Cantor manifolds not embeddable into each other.

On hereditarily indecomposable continua, Henderson compacta and a question of Yohe

We answer a question of Yohe by showing that there exists a family of continuum many topologically different hereditarily indecomposable Cantor manifolds without any non-trivial weakly infinite-dimensional subcontinua. This family may consist either of compacta containing one-dimensional subsets or of compacta containing no weakly infinite-dimensional subsets of positive dimension.

Continuum many Frechet types of hereditarily strongly infinite-dimensional Cantor manifolds

In this note we construct a family of continuum many hereditarily strongly infinite-dimensional Cantor manifolds such that for every two spaces from this family, no open subset of one is embeddable into the other.

Remark on compactifications of metrizable spaces

Abstract We show that for a given metrizable separable space X and a family{fi};∞i=1 of continuous functions of X into X the set of all continuous embeddings h of the space X into the Hilbert cube Iω such that all functions hfih-1 are extendable to continuous functions f i : h(X)) → h(X) contains a dense Gδ-set in the function space C(X,Iω). This strengthens and gives a new proof of some earlier results of Engelking, de Groot and McDowell.

Function spaces in dimension theory and factorization theorems

Abstract The main result of this paper in the following theorem: given a mapping f:X→Z of a Tychonoff space X into a metrizable space Z of weight τ, for almost every (in the sense of Baire category) mapping g:X → J(τ)ω into the countable power of the hedgehog space of spininess τ we have 1. (i) dim g(X) ⩽ dim X and 2. (ii) there exists a mapping h:g(X) → Z satisfying f = h ∘ g. This gives a new approach to factorization theorems of Pasynkov. We show also that, given a mapping f:X → X of a metrizable space X of weight τ into itself, for almost every mapping h:X → J(τ)ω there exists u:J(τ)ω → J(τ)ω such that u ∘ h = h ∘ ƒ.

On J. Nagata's universal spaces

Abstract The main result of this paper is the following extension of an embedding theorem by Nagata: given a sequence of zero-dimensional sets X 1 , X 2 ,… in a metrizable space X of weight τ⩾ℵ 0 , the set of homeomorphic embeddings h of X into S (τ) ℵ 0 , satisfying h ( X n )⊂ K n -1 ( τ ) for n = 1,2,…, is dense in the function space of all continuous mappings of X into S (τ) ℵ 0 , where K n (τ) is the n -dimensional universal Nagatas space in the countable product of the star-space S (τ) of weight τ. This seems to be a new result even in the separable case τ=χ 0 and provides in particular an answer to a question asked by Kuratowski (see Remark 2.6 for the details).

A few remarks concerning countable unions of finite-dimensional spaces

Abstract We use Nagatas universal spaces and function space methods to give an alternative proof of recent theorems of Engelking concerning strongly countable-dimensional and locally finite-dimensional spaces.