Criterion of completeness of topological linear spaces and topological Abelian groups

A criterion of completeness, established earlier by the author for locally convex spaces, is generalized to arbitrary topological linear spaces and topological Abelian groups.     

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.     

Compact Hausdorff fuzzy topological spaces are topological

Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces.     

Edelstein's fixed point theorem in topological spaces

An extension to topological spaces of a wellknown fixed point theorem of M. Edelstein for contractive mappings on metric spaces is presented. Results based on the generalized Edelstein's theorem are also established concerning the existence of fixed points of continuous selfmaps on a topological space. As a special case a compact starshaped subset of a linear topological space is considered. The results extend the fixed point theoremsfor nonexpansive mappings on a compact metric space of L.F.Guseman, Jr. and B.C. Peters, Jr.     

Connectedness in topological linear spaces

The following properties, well known for normed linear spaces of dimension ≧2, are established for an arbitrary topological linear space of dimension ≧2: (a) every neighborhood of 0 contains one whose complement is connected; (b) the complement of a bounded set has exactly one unbounded component. Research supported by the National Science Foundation, U.S.A. (NSF-GP-378).     

A note on the topological transversality theorem for acyclic maps

A new topological transversality theorem is presented for acyclic maps. The analysis relies on Urysohn’s Lemma and the fact that the unit sphere is contractible in infinite dimensional normed linear spaces.     

On L-fuzzy topological spaces

L-fuzzy sets and, subsequently, L-fuzzy topological spaces are considered in this paper, where L is a complete and completely distributive non-atomic Boolean algebra. A few separation properties as well as subspace L-fuzzy topology are defined here, in the light of an earlier paper. These are more similar to ordinary topological spaces than to fuzzy topological spaces.     

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.     

Realization of cartesian closed topological hulls

The cartesian closed topological (CCT) hull of a concrete categoryK is the smallest full CCT extension ofK. A general method for describing the CCT hulls is presented and applied to render unified proofs for the basic examples ofK: topological spaces, uniform spaces, pretopological spaces, compact T₂ spaces, metrizable spaces, and completely regular spaces.     

Summability results for operator matrices on topological vector spaces

Basic summability results are established for matrices of linear and some nonlinear mappings between topological vector spaces.     

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.     

Supercomplete topological spaces

Supercomplete topological spaces and other variants of supercompleteness are defined in this paper. The main idea is to give different characterizations of Čech complete spaces. In particular it is shown that for paracompacta (and so for metrizable spaces) and topological groups the two notions of supercompleteness and Čech completeness coincide.      

Polynomials in topological vector spaces over valued fields

We consider the vector space of continuousm-homogeneous polynomials between topological vector spaces over a non-trivially valued field of characteristic zero and certain natural vector topologies on such spaces, and we prove polynomial versions of certain well known theorems of the linear theory of locally convex spaces. Partially supported by CNPq.     

C-scattered fuzzy topological spaces

In this paper, we define the concept of C-scattered fuzzy topological spaces and obtain some properties about them. In particular, we study the relation between C-scattered spaces and its fuzzy extension, it is proved that C-scattered fuzzy topological spaces are invariant by fuzzy perfect maps, and that, in the realm of paracompact fuzzy topological spaces, the C-scattered spaces verify that their product by other fuzzy spaces is also paracompact fuzzy.     

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.     

Set-valued topological contractions

By introducing a new concept called “set-valued topological contraction” in topological spaces, the existence and uniqueness of endpoints for both set-valued and single-valued topological contractions have been established.     

Transitivities of maps of general topological spaces

In this paper we investigate orbit-transitivity, strong orbit-transitivity, ω-transitivity and open-set-transitivity of maps of general topological spaces. The relation between these transitivities is studied. We discuss various topological spaces, containing pseudo-regular spaces, partially completable spaces, and topological spaces without quasi-isolated points. Several conditions on spaces and on continuity for one transitivity to imply another transitivity are given.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

A nonalgebraic approach to the theory of linear topological spaces

The basic theory developed in the setting of an algebraic system endowed with a topology which renders the operations continuous (e.g. a topological linear space) can be produced entirely by geometric (set-theoretic) notions without using algebra. This theory affords an interesting synthesis of classical results, and is more general. For example, our theory applies to a larger class of spaces than linear topological spaces, but is powerful enough to prove the appropriate formulation in this setting of the Krein-Milman theorem on extreme points. This theorem is the goal of this particular article, as an example of the direction which the theory can be made to take.