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On topological classification of non-archimedean Fréchet spaces

Wiesƚaw Śliwa (2004)

Czechoslovak Mathematical Journal

We prove that any infinite-dimensional non-archimedean Fréchet space E is homeomorphic to D where D is a discrete space with c a r d ( D ) = d e n s ( E ) . It follows that infinite-dimensional non-archimedean Fréchet spaces E and F are homeomorphic if and only if d e n s ( E ) = d e n s ( F ) . In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field 𝕂 is homeomorphic to the non-archimedean Fréchet space 𝕂 .

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, U α : α , such that U α U β whenever β ≤ α for all α , β . 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 this result...

On vector measures

Corneliu Constantinescu (1975)

Annales de l'institut Fourier

Let be the Banach space of real measures on a σ -ring R , let ' be its dual, let E be a quasi-complete locally convex space, let E ' be its dual, and let μ be an E -valued measure on R . If is shown that for any θ ' there exists an element θ d μ of E such that x ' μ , θ = θ d μ , x ' for any x ' E ' and that the map θ θ d μ : ' E is order continuous. It follows that the closed convex hull of μ ( R ) is weakly compact.

On vector spaces and algebras with maximal locally pseudoconvex topologies

A. Kokk, W. Żelazko (1995)

Studia Mathematica

Let X be a real or complex vector space. We show that the maximal p-convex topology makes X a complete Hausdorff topological vector space. If X has an uncountable dimension, then different p give different topologies. However, if the dimension of X is at most countable, then all these topologies coincide. This leads to an example of a complete locally pseudoconvex space X that is not locally convex, but all of whose separable subspaces are locally convex. We apply these results to topological algebras,...

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