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Addition theorems, D -spaces and dually discrete spaces

Ofelia Teresa Alas, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2009)

Commentationes Mathematicae Universitatis Carolinae

A neighbourhood assignment in a space X is a family 𝒪 = { O x : x ∈ X } of open subsets of X such that x ∈ O x for any x ∈ X . A set Y ⊆ X is a kernel of 𝒪 if 𝒪 ( Y ) = ⋃ { O x : x ∈ Y } = X . If every neighbourhood assignment in X has a closed and discrete (respectively, discrete) kernel, then X is said to be a D -space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf P -space is a D -space and we prove an addition...

Addition theorems for dense subspaces

Aleksander V. Arhangel'skii (2015)

Commentationes Mathematicae Universitatis Carolinae

We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space X which is the union of two dense metrizable subspaces need not be a p -space. However, if a normal space X is the union of a finite family μ of dense subspaces each of which is metrizable by a complete metric, then X is also metrizable by...

Addresses

(1979)

Abstracta. 7th Winter School on Abstract Analysis

Adequate Compacta which are Gul’ko or Talagrand

Čížek, Petr, Fabian, Marián (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 54H05, 03E15, 46B26We answer positively a question raised by S. Argyros: Given any coanalytic, nonalytic subset Σ′ of the irrationals, we construct, in Mercourakis space c1(Σ′), an adequate compact which is Gul’ko and not Talagrand. Further, given any Borel, non Fσ subset Σ′ of the irrationals, we construct, in c1(Σ′), an adequate compact which is Talagrand and not Eberlein.Supported by grants AV CR 101-90-03, and GA CR 201-01-1198

Admissible maps, intersection results, coincidence theorems

Mircea Balaj (2001)

Commentationes Mathematicae Universitatis Carolinae

We obtain generalizations of the Fan's matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.

Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan, Natalia Jonard-Pérez (2013)

Fundamenta Mathematicae

For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ) is homeomorphic...

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