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On generalized topological spaces I

Artur Piękosz (2013)

Annales Polonici Mathematici

We begin a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings. We reformulate the axioms. Generalized topology is found to be connected with the concept of a bornological universe. Both GTS and its full subcategory SS of small spaces are topological categories. The second part of this paper will also appear in this journal.

On generalized topological spaces II

Artur Piękosz (2013)

Annales Polonici Mathematici

This is the second part of A. Piękosz [Ann. Polon. Math. 107 (2013), 217-241]. The categories GTS(M), with M a non-empty set, are shown to be topological. Several related categories are proved to be finitely complete. Locally small and nice weakly small spaces can be described using certain sublattices of power sets. Some important elements of the theory of locally definable and weakly definable spaces are reconstructed in a wide context of structures with topologies.

On nonbornological barrelled spaces

Manuel Valdivia (1972)

Annales de l'institut Fourier

If E is the topological product of a non-countable family of barrelled spaces of non-nulle dimension, there exists an infinite number of non-bornological barrelled subspaces of E . The same result is obtained replacing “barrelled” by “quasi-barrelled”.

Reflexivity of spaces of weakly summable sequences.

L. Oubbi, M. A. Ould Sidaty (2007)

RACSAM

We deal with the space of Λ-summable sequences from a locally convex space E, where Λ is a metrizable perfect sequence space. We give a characterization of the reflexivity of Λ(E) in terms of that of Λ and E and the AK property. In particular, we prove that if Λ is an echelon sequence space and E is a Fréchet space then Λ(E) is reflexive if and only if Λ and E are reflexive.

Some examples on quasi-barrelled spaces

Manuel Valdivia (1972)

Annales de l'institut Fourier

The three following examples are given: a bornological space containing a subspace of infinite countable codimension which is not quasi-barrelled, a quasi-barrelled 𝒟 -space containing a subspace of infinite countable codimension which is not 𝒟 -space, and bornological barrelled space which is not inductive limit of Baire space.

Some properties of the tensor product of Schwartz εb-spaces.

Belmesnaoui Aqzzouz, M. Hassan el Alj, Redouane Nouira (2007)

RACSAM

We define the ε-product of an εb-space by quotient bornological spaces and we show that if G is a Schwartz εb-space and E|F is a quotient bornological space, then their εc-product Gεc(E|F) defined in [2] is isomorphic to the quotient bornological space (GεE)|(GεF).

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