Algebraic objects generated by topological structure

The survey examines the properties of the Tikhonov space X that depend on the topological properties of the function space C(X) with pointwise topology and of the free topological group F(X). It covers the part of the theory of topological groups dealing with topological invariants and topological classification of topological groups. Continuous images of topological groups are considered in the context of studying the actions of topological groups on spaces.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 141–198, 1987.     

On Chains in <Emphasis Type="BoldItalic">H</Emphasis>-Closed Topological Pospaces

We study chains in an H-closed topological partially ordered space. We give sufficient conditions for a maximal chain L in an H-closed topological partially ordered space (H-closed topological semilattice) under which L contains a maximal (minimal) element. We also give sufficient conditions for a linearly ordered topological partially ordered space to be H-closed. We prove that a linearly ordered H-closed topological semilattice is an H-closed topological pospace and show that in general, this is not true. We construct an example of an H-closed topological pospace with a non-H-closed maximal chain and give sufficient conditions under which a maximal chain of an H-closed topological pospace is an H-closed topological pospace.     

On the Brandt λ 0-extensions of monoids with zero

We study algebraic properties of the Brandt λ ⁰-extensions of monoids with zero and non-trivial homomorphisms between the Brandt λ ⁰-extensions of monoids with zero. We introduce finite, compact topological Brandt λ ⁰-extensions of topological semigroups and countably compact topological Brandt λ ⁰-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt λ ⁰-extensions of topological monoids with zero. We also describe a category whose objects are ingredients in the constructions of finite (compact, countably compact) topological Brandt λ ⁰-extensions of topological monoids with zeros.     

Topological Loewner theory on Riemann surfaces

We prove that topological evolution families on a Riemann surface S are rather trivial unless S is conformally equivalent to the unit disc or the punctuated unit disc. We also prove that, except for the torus where there is no non-trivial continuous Loewner chain, there is a topological evolution family associated to any topological Loewner chain and, conversely, any topological evolution family comes from a topological Loewner chain on the same Riemann surface.     

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.     

Embedding topological semigroups in topological groups

In [6] Rothman investigated the problem of embedding a topological semigroup in a topological group. He defined a concept calledProperty F and showed that Property F is a necessary and sufficient condition for embedding a commutative, cancellative topological semigroup in its group of quotients as an open subset. This paper announces a generalization of Rothman’s result by definingProperty E and stating that a completely regular topological semigroup S can be embedded in a topological group by a topological isomorphism if and only if S can be embedded (algebraically) in a group and S has Property E. Property E is defined by first constructing a free topological semigroup (Theorem 1.1). This construction resembles the one in [4] for a free topological group. Full details, examples, and other embedding results will appear elsewhere. Some of the results in this paper were contained in the author’s doctoral dissertation written at Rutgers University under Professor Louis F. McAuley.     

关于几乎拓扑群的注记

In this paper, we mainly discuss some generalized metric properties and the cardinal invariants of almost topological groups. We give a characterization for an almost topological group to be a topological group and show that:(1) Each almost topological group that is of countable π-character is submetrizable;(2) Each left λ-narrow almost topological group isλ-narrow;(3) Each separable almost topological group is ω-narrow. Some questions are posed.     

Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem

In this paper we prove the semialgebraic version of Palais' covering homotopy theorem, and use this to prove Bredon's covering mapping cylinder conjecture positively in the semialgebraic category. Bredon's conjecture was originally stated in the topological category, and a topological version of our semialgebraic proof of the conjecture answers the original topological conjecture for topological G-spaces over “simplicial” mapping cylinders.     

Fibrations of topological stacks

In this note we define fibrations of topological stacks and establish their main properties. When restricted to topological spaces, our notion of fibration coincides with the classical one. We prove various standard results about fibrations (long exact sequence for homotopy groups, Leray–Serre and Eilenberg–Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. As an immediate consequence of the machinery we develop, we also prove van Kampen?s theorem for fundamental groups of topological stacks.     

Quasi-uniformities on topological semigroups and bicompletion

We study the bicompletion of the quasi-uniformities that are induced in a natural way on a topological semigroup which has a neutral element. In particular, we show that if X is a topological semigroup, with neutral element, for which the left translations are open, then the bicompletion of the left quasi-uniformity of X can be considered a topological semigroup which contains the topological space X as a sup-dense subsemigroup. The bicompletion in the case that the left translations are not necessarily open is also discussed. In particular, both Abelian and left-cancellable topological semigroups are considered. For semigroups which are (left-)cancellable or which are locally totally bounded, theorems similar to those known from the classical theory of (para)topological groups are established. July 1, 1999     

Higher topological cyclic homology and the Segal conjecture for tori

We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory.The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology.We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus.     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

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.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

Functorial Duality for Ortholattices and De Morgan Lattices

Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.     

Unique Nearness Structures

We investigate which symmetric topological spaces have precisely one compatible nearness structure. We show how this situation exists for T ₁ topological spaces and for T ₂ topological spaces. The compatible nearness structures on a symmetric topological space form a sort of “interval” of structures with the topology itself as the topmost.     

Fréchet-Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan S_ω in a topological group G admitting a functorial embedding [0,1]⊂G. The latter means that each autohomeomorphism of [0,1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing S_ω.     

On intuitionistic modal and tense logics and their classical companion logics: Topological semantics and bisimulations

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics.