Non-productive duality properties of topological groups

Productivity of sequences with respect to a given weight function

Given a function f: N --> (omega+1)-{0}, we say that a faithfully indexed sequence {a_n: n in N} of elements of a topological group G is: (i) f-Cauchy productive (f-productive) provided that the sequence {prod_{n=0}^m a_n^{z(n)}: m in N} is left Cauchy (converges to some element of G, respectively) for each function z: N --> Z such that |z(n)| <= f(n) for every n in N; (ii) unconditionally f-Cauchy productive (unconditionally f-productive) provided that the sequence {a_{s(n)}: n in N\} is (f\circ s)-Cauchy productive (respectively, (f\circ s)-productive) for every bijection s: N --> N. (Biject…

Duality of $κ$-normed topological vector spaces and their applications

A duality of $κ$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $κ$-normability of a topological vector space implies its local convexity. There are given applications of $κ$-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economy.

序拓扑空间的石对偶

Stone Duality for Preordered Topological Spaces

本文提出序拓扑空间的石对偶理论，基于二拓扑空间和序集的对偶思想，构建新的对偶关系。

关于阿贝尔拓扑群 Pontryagin 对偶性的若干注记

Some notes on Pontryagin duality of abelian topological groups

本文探讨了阿贝尔pro-Lie群中Pontryagin对偶性相关的若干问题，分析了该领域内的核心方法与主要贡献。

Distributive envelopes and topological duality for lattices via canonical extensions

We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a se…

Pontryagin duality between compact and discrete abelian inverse monoids

For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify …

Stone duality for topological theories

In the context of categorical topology, more precisely that of T-categories [Hofmann, 2007], we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of "Stone duality", and show that Cauchy completeness of a T-category is precisely its sobriety.

Complete Regularity: Kopperman's duality {\it à la quantale}

Nearly three decades from his celebrated result, we study a modern refinement and strengthening of Kopperman's full metrisabilty of all topological spaces. Within this new theory of \emph{V-spaces}, developed by Flagg and Weiss, we investigate several topological notions and their metric counterpart. Among our main results is the reconstruction, in terms of V-spaces, of Kopperman's equivalence between symmetric value semigroups and completely regular topologies. We conclude our work by revisiting some classical topological results and their almost evident validity through this metric lens.

Topological T-duality for Stacks using a Gysin Sequence

In this paper we study the topological T-dual of spaces with a non-free circle action mainly using the stack theory method of Bunke and co-workers \cite{Bunke1}. We first compare three formalisms for obtaining the Topological T-dual of a semi-free $S^1$-space in a simple example. Then, we calculate the T-dual of general KK-monopole backgrounds using the stack theory method. We define the dyonic coordinate for these backgrounds. We introduce an approach to Topological T-duality using classifying spaces which simultaneously generalizes the methods of Bunke et al \cite{Bunke1} and Mathai and Wu \…

A four for the price of one duality principle for distributive spaces

In this paper we consider topological spaces as generalised orders and characterise those spaces which satisfy a (suitably defined) topological distributive law. Furthermore, we show that the category of these spaces is dually equivalent to a certain category of frames by simply observing that both sides represent the idempotents split completion of the same category.

完全不连通（非度量）Gelfand对偶

Totally Disconnected (non-metric) Gelfand Duality

本文推广Gelfand对偶至完全不连通拓扑域，刻画可表示为紧致空间连续函数代数的代数，并建立紧致F-Tychonoff空间与交换F-代数间的对偶伴随，方法不依赖解析工具。

Stone duality, topological algebra, and recognition

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognizable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in compute…

Topological Representation of Precontact Algebras and a Connected Version of the Stone Duality Theorem -- I

The notions of a {\em 2-precontact space}\/ and a {\em 2-contact space}\/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem for a kind of compact $T_0$-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion …