Displaying similar documents to “ M -barrelled spaces, M -bornological spaces: Ad addendum”

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces ( X , τ X ) and ( Y , τ Y ) are called T₁-complementary provided that there exists a bijection f: X → Y such that τ X and f - 1 ( U ) : U ∈ τ Y are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size 2 which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

When ( E , σ ( E , E ' ) ) is a D F -space?

Dorota Krassowska, Wiesƚaw Śliwa (1992)

Commentationes Mathematicae Universitatis Carolinae

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Let ( E , t ) be a Hausdorff locally convex space. Either ( E , σ ( E , E ' ) ) or ( E ' , σ ( E ' , E ) ) is a D F -space iff E is of finite dimension (THEOREM). This is the most general solution of the problem studied by Iyahen [2] and Radenovič [3].

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, U α : α ∈ ℕ ℕ , such that U α ⊂ U β whenever β ≤ α for all α , β ∈ ℕ ℕ . The class of all metrizable topological groups is a proper subclass of the class T G of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group G ∈ T G is metrizable, and hence G is strictly angelic. We deduce from...

Sequential closures of σ -subalgebras for a vector measure

Werner J. Ricker (1996)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a locally convex space, m : Σ → X be a vector measure defined on a σ -algebra Σ , and L 1 ( m ) be the associated (locally convex) space of m -integrable functions. Let Σ ( m ) denote { χ E ; E ∈ Σ } , equipped with the relative topology from L 1 ( m ) . For a subalgebra 𝒜 ⊆ Σ , let 𝒜 σ denote the generated σ -algebra and 𝒜 ¯ s denote the closure of χ ( 𝒜 ) = { χ E ; E ∈ 𝒜 } in L 1 ( m ) . Sets of the form 𝒜 ¯ s arise in criteria determining separability of L 1 ( m ) ; see [6]. We consider some natural questions concerning 𝒜 ¯ s and, in particular, its relation to χ ( 𝒜 σ ) . It is shown that...