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Some remarks on Baire-like spaces

100%

Kąkol J.

Commentationes Mathematicae

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1981

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tom 22

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nr 2

EN

The article contains no abstract

2

Countable codimensional subspaces of spaces with topologies determined by a family of balanced sets

100%

Kąkol J.

Commentationes Mathematicae

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1983

|

tom 23

|

nr 2

EN

The article contains no abstract

3

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

51%

Gabriyelyan S., Kąkol J., Plebanek G.

Studia Mathematica

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2016

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tom 233

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nr 2

119-139

EN

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset 𝒦 of $C_{k}(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli iff $C_{k}(X)$ is a $k_{ℝ}$-space iff X is locally compact. Moreover, $C_{k}(X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of ℓ₁, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ₁, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_{ℝ}$-space, (iv) $B_{w}$ is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ₁ iff each closed and convex bounded subset of F is Ascoli in the weak topology. Moreover we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff F is either finite-dimensional or isomorphic to $𝕂^{ℕ}$, where 𝕂 ∈ {ℝ,ℂ}.

Ograniczanie wyników

2 Commentationes Mathematicae

1 Studia Mathematica

3 Kąkol J.

1 Gabriyelyan S.

1 Plebanek G.

1 2016

1 1983

1 1981

Some remarks on Baire-like spaces

Countable codimensional subspaces of spaces with topologies determined by a family of balanced sets

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces