On properties of h-homogeneous spaces of first category

Abstract

A metric space X is called h-homogeneous if Ind X = 0 and each nonempty open-closed subset of X is homeomorphic to X. We describe how to assign an h-homogeneous space of first category and of weight k to any strongly zero-dimensional metric space of weight k. We investigate the properties of such spaces. We show that if Q is the space of rational numbers and Y is a strongly zero-dimensional metric spaceKeldysh proved that any two canonical elements of the Borel class α are homeomorphic. The last theorem is generalized for the nonseparable case.All topological spaces under discussion are metrizable and strongly zero-dimensional (i.e., Ind X = 0).The aim of this paper is to describe how to assign an h-homogeneous space h( X, k) of first category and of weight k to any strongly zero-dimensional metric space X of weight k. Using this construction, we generalize for the nonseparable case the L. Keldysh theorem [4] stating that any two canonical elements of class α are homeomorphic.Let X be a space of weight k. We shall show (see Theorems 6 and 8) that the family H k (X) = {Y : w(Y ) = k, Y is an h-homogeneous space of first category, and Y contains a closed copy of X} has a unique (up to homeomorphism) element h( X, k) such that every Y ∈ H k (X) contains a closed copy of h( X, k). This element is called the h-homogeneous extension of the space X of weight k with respect to spaces of first category. Briefly, h( X, k) is the h-homogeneous extension of X . The proof of Theorem 6 gives a direct construction of the space h( X, k) for any space X .Theorem 7 states that if X is an h-homogeneous space of weight k, then h( X, k) ≈ Q × X . This case is the simplest. A topological characterization of Q × X for an h-homogeneous space X was studied by van Mill [9] (in the separable case) and by Ostrovsky [10]. For an arbitrary space X the product Q × X might not be h-homogeneous.Theorem 4 implies that every h-homogeneous space X of first category is homeomorphic to the product Q × X . This result suggests the following Question. Let X be an h-homogeneous space of first category and Q be the space of rational numbers. Is there an h-homogeneous space Y such that X is homeomorphic to Q × Y and Y is not of first category?Theorem 10 shows that h( X, k) is homeomorphic to h(Y , k) if and only if X ∈ σ LF(Y ) and Y ∈ σ LF( X). We also obtain (see Theorem 11) that the Cartesian product of the h-homogeneous extensions of two spaces is homeomorphic to the h-homogeneous extension of the product of these spaces.Let us recall the L. Keldysh definition [4] of a canonical element of the Borel class α, where α is a countable ordinal. Put M α = {X ⊆ B(ω): X is of multiplicative class α}. A set U ∈ M α is called universal for M α if for each X ∈ M α there

Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

“…Knaster-Reichbach covers were introduced in [30] and have been successfully applied by several authors, including van Engelen, Medvedev and Ostrovskiȋ. Let us mention for example the articles [5], [22], [23], [24], [25] and [34], where one can find much more general results than the ones stated here. The first application of this technique to the theory of countable dense homogeneity was recently given by Hernández-Gutiérrez, Hrušák and van Mill in [10].…”

σ-Homogeneity of Borel sets

“…Note that S. Medvedev proved that every h-homogeneous space Y ⊂ C of the first category in itself is homeomorphic to Y × Q. For more details we refer the reader to[6].…”