Sequential convergence in topological vector spaces

For a given linear topology , on a vector spaceE, the finest linear topology having the same convergent sequences, and the finest linear topology onE having the same precompact sets, are investigated. Also, the sequentially bornological spaces and the sequentially barreled spaces are introduced and some of their properties are studied.     

Topological characteristics of free bases of topological groups and topological vector spaces

Mutual dimensional properties of basis-equivalent (b-equivalent) and weakly l-equivalent topological spaces are studied. It is shown that the b-equivalence does not preserve bicompactness; in particular, b-equivalent topological spaces can be non-l-equivalent. Bibliography: 18 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 76–87. Translated by O. A. Ivanov.     

Fuzzy topological vector spaces I

Some of the properties of fuzzy topological vector spaces are investigated. Also, there are given necessary and sufficient conditions for a family of fuzzy sets, in a vector space E, to be the family of all neighborhoods of zero for a fuzzy linear topology.     

Fuzzy topological vector spaces II

Some of the properties of the fuzzy seminormed and fuzzy normed spaces are studied. Also, the notion of a bornological fuzzy linear space is given and some of the properties of such a space are investigated.     

CLP-compactness for topological spaces and groups

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct:

(i)

a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and

(ii)

a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G→H, with a Hausdorff group H, are open).

     

Bourbaki spaces of topological groups

A study is presented of the relationship between the topological and uniformity properties of a group G and the spaces (G), (G) of all nonempty closed subsets and closed subgroups of G. A base for the neighborhood system of a closed subset X of G is formed by the sets S(X, U)={Y Y XU, X YU}, where U ranges over all neighborhoods of the identity in G. Criteria are obtained for the space (G) and some of its subspaces to be totally bounded and locally totally bounded. Some classes of groups with compact spaces (G) are described. It is proved that the spaces (G), (G) are complete in the case of projective metrizable groups G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 542–549, April, 1990.     

Semitopological groups,Bouziad spaces and topological groups

A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we use topological games to show that many semitopological groups are in fact topological groups.     

Theory of regularizability in topological vector spaces

Two known definitions of regularizability for topological vector spaces are found to be equivalent. Regularizability in the sense of Tikhonov is considered in reflexive linear metric spaces. In particular, an example is presented of a linear continuous injective operator on a reflexive Frécnet space whose inverse cannot be regularized. The latter indicates the sharp difference between regularizability in Fréchet spaces and in Banach spaces, respectively.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 777–781, June, 1990.     

Operator matrices on topological vector spaces

A series of basic summability results are established for matrices of linear and some nonlinear operators between topological vector spaces.