We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2 ω , then X has c types of countable dense subsets. We suggest a generalization of the λ-set for non-separable spaces. Let X be an h-homogeneous Λ-set. Then X is densely homogeneous and X \ A is homeomorphic to X for every σ-discrete subset A ⊂ X.2010 Mathematics Subject Classification. 54H05, 54E52.
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
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