Displaying similar documents to “Definable hereditary families in the projective hierarchy”

Generalized projections of Borel and analytic sets

Marek Balcerzak (1996)

Colloquium Mathematicae

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For a σ-ideal I of sets in a Polish space X and for A ⊆ X 2 , we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where A x =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a ∑ 2 0 -supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [ ∑ 1 1 ( X 2 ) ] = ∑ 1 1 ( X ) for a wide class of ∑ 2 0 -supported σ-ideals.

Functions characterized by images of sets

Krzysztof Ciesielski, Dikran Dikrajan, Stephen Watson (1998)

Colloquium Mathematicae

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For non-empty topological spaces X and Y and arbitrary families 𝒜 ⊆ 𝒫 ( X ) and ℬ ⊆ 𝒫 ( Y ) we put 𝒞 𝒜 , ℬ =f ∈ Y X : (∀ A ∈ 𝒜 )(f[A] ∈ ℬ ) . We examine which classes of functions ℱ ⊆ Y X can be represented as 𝒞 𝒜 , ℬ . We are mainly interested in the case when ℱ = 𝒞 ( X , Y ) is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class ℱ = 𝒞 (X,ℝ) is not equal to 𝒞 𝒜 , ℬ for any 𝒜 ⊆ 𝒫 ( X ) and ℬ ⊆ 𝒫 (ℝ). Thus, 𝒞 (X,ℝ) cannot be characterized by images of sets. We also show that none of the following...

Forcing countable networks for spaces satisfying R ( X ω ) = ω

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (1996)

Commentationes Mathematicae Universitatis Carolinae

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We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that in V P X is a countable union of 0 -dimensional subspaces of countable weight. We also show that this theorem is sharp in two different senses: (i) we cannot get rid of using generic extensions, (ii) we have to consider all finite powers of X .

Embedding of the ordinal segment [ 0 , ω 1 ] into continuous images of Valdivia compacta

Ondřej F. K. Kalenda (1999)

Commentationes Mathematicae Universitatis Carolinae

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We prove in particular that a continuous image of a Valdivia compact space is Corson provided it contains no homeomorphic copy of the ordinal segment [ 0 , ω 1 ] . This generalizes a result of R. Deville and G. Godefroy who proved it for Valdivia compact spaces. We give also a refinement of their result which yields a pointwise version of retractions on a Valdivia compact space.

Kneser’s theorem for upper Banach density

Prerna Bihani, Renling Jin (2006)

Journal de Théorie des Nombres de Bordeaux

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Suppose A is a set of non-negative integers with upper Banach density α (see definition below) and the upper Banach density of A + A is less than 2 α . We characterize the structure of A + A by showing the following: There is a positive integer g and a set W , which is the union of ⌈ 2 α g - 1 ⌉ arithmetic sequences [We call a set of the form a + d ℕ an arithmetic sequence of difference d and call a set of the form { a , a + d , a + 2 d , ... , a + k d } an arithmetic progression of difference d . So an arithmetic progression is finite and an arithmetic...

Examples of functions -extendable for each finite, but not ∞ -extendable

Wiesław Pawłucki (1998)

Banach Center Publications

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In Example 1, we describe a subset X of the plane and a function on X which has a k -extension to the whole ℝ 2 for each finite, but has no ∞ -extension to ℝ 2 . In Example 2, we construct a similar example of a subanalytic subset of ℝ 5 ; much more sophisticated than the first one. The dimensions given here are smallest possible.