Displaying similar documents to “Addition theorems, D -spaces and dually discrete spaces”

A quest for nice kernels of neighbourhood assignments

Raushan Z. Buzyakova, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2007)

Commentationes Mathematicae Universitatis Carolinae

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Given a topological property (or a class) 𝒫 , the class 𝒫 * dual to 𝒫 (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment { O x : x ∈ X } there is Y ⊂ X with Y ∈ 𝒫 and ⋃ { O x : x ∈ Y } = X . The spaces from 𝒫 * are called . We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define D -spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...

Diagonals and discrete subsets of squares

Dennis Burke, Vladimir Vladimirovich Tkachuk (2013)

Commentationes Mathematicae Universitatis Carolinae

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In 2008 Juhász and Szentmiklóssy established that for every compact space X there exists a discrete D ⊂ X × X with | D | = d ( X ) . We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf Σ -space X and hence X ω is d -separable. We give an example of a countably compact space X such that X ω is not d -separable. On the other hand, we show that for any Lindelöf p -space X there exists a discrete subset D ⊂ X × X such that Δ = { ( x , x ) : x ∈ X } ⊂ D ¯ ; in particular, the diagonal Δ is a retract of...

On some classes of spaces with the D -property

Juan Carlos Martínez (2014)

Commentationes Mathematicae Universitatis Carolinae

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We shall prove that under CH every regular meta-Lindelöf P -space which is locally D has the D -property. In addition, we shall prove that a regular submeta-Lindelöf P -space is D if it is locally D and has locally extent at most ω 1 . Moreover, these results can be extended from the class of locally D -spaces to the wider class of 𝔻 -scattered spaces. Also, we shall give a direct proof (without using topological games) of the result shown by Peng [On spaces which are D, linearly D and transitively...