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The Banach–Mazur game and σ-porosity

Miroslav Zelený (1996)

Fundamenta Mathematicae

It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

The minimum uniform compactification of a metric space

R. Grant Woods (1995)

Fundamenta Mathematicae

It is shown that associated with each metric space (X,d) there is a compactification u d X of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of u d X are presented, and a detailed study of the structure of u d X is undertaken. This culminates in a topological characterization of the outgrowth u d ℝ n ∖ ℝ n , where ( ℝ n , d ) is Euclidean n-space with its usual metric.

The Niemytzki plane is ϰ -metrizable

Wojciech Bielas, Andrzej Kucharski, Szymon Plewik (2021)

Mathematica Bohemica

We prove that the Niemytzki plane is ϰ -metrizable and we try to explain the differences between the concepts of a stratifiable space and a ϰ -metrizable space. Also, we give a characterisation of ϰ -metrizable spaces which is modelled on the version described by Chigogidze.

The positive cone of a Banach lattice. Coincidence of topologies and metrizability

Zbigniew Lipecki (2023)

Commentationes Mathematicae Universitatis Carolinae

Let X be a Banach lattice, and denote by X + its positive cone. The weak topology on X + is metrizable if and only if it coincides with the strong topology if and only if X is Banach-lattice isomorphic to l 1 ( Γ ) for a set Γ . The weak * topology on X + * is metrizable if and only if X is Banach-lattice isomorphic to a C ( K ) -space, where K is a metrizable compact space.

The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces

M. Holmes (1992)

Fundamenta Mathematicae

This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0,1] of functions from [0,1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0,1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry....

Topological spaces admitting a unique fractal structure

Christoph Bandt, T. Retta (1992)

Fundamenta Mathematicae

Each homeomorphism from the n-dimensional Sierpiński gasket into itself is a similarity map with respect to the usual metrization. Moreover, the topology of this space determines a kind of Haar measure and a canonical metric. We study spaces with similar properties. It turns out that in many cases, "fractal structure" is not a metric but a topological phenomenon.

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