Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. Both of them imply easily the Tarski Duality Theorem, as well as two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.
We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to étale groupoids, obtaining a duality with an elementary class of inverse semigroups.
Gelfand-Naimark-Stone duality establishes a dual equivalence between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category ${\boldsymbol{\mathit{uba}\ell}}$ of uniformly complete bounded archimedean $\ell$-algebras. We extend this duality to the category ${\sf CReg}$ of completely regular spaces. This we do by first introducing basic extensions of bounded archimedean $\ell$-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category ${\boldsymbol{\mathit{ubasic}}}$ of uniformly complete basic extensions and the category ${\sf C}$ of comp…
In [G. Dimov and E. Ivanova-Dimova, Two extensions of the Stone Duality to the category of zero-dimensional Hausdorff spaces, arXiv:1901.04537v4, 1--33], extending the Stone Duality Theorem, we proved two duality theorems for the category ZDHaus of zero-dimensional Hausdorff spaces and continuous maps. Now we derive from them the extension of the Stone Duality Theorem to the category BooleSp of zero-dimensional locally compact Hausdorff spaces and continuous maps obtained in [G. Dimov, Some generalizations of the Stone Duality Theorem, Publicationes Mathematicae Debrecen, 80 (2012), 255--293],…
We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting the decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of b…
Gelfand-Naimark-Stone duality provides an algebraic counterpart of compact Hausdorff spaces in the form of uniformly complete bounded archimedean $\ell$-algebras. In [4] we extended this duality to completely regular spaces. In this article we use this extension to characterize normal, Lindëlof, and locally compact Hausdorff spaces. Our approach gives a different perspective on the classical theorems of Katětov-Tong and Stone-Weierstrass.
A condensed set is a sheaf on the site of Stone spaces and continuous maps. We prove that condensed sets are equivalent to sheaves on the site of compact Hausdorff spaces and continuous maps. As an application, we show that there exists a model structure on the category of condensed sets.
Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category $\bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $\bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is pres…
We prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in arXiv:1508.02220v3, 1-44 (to appear in Topology Appl.), the recent duality theorems of Bezhanishvili, G., Bezhanishvili, N., Sourabh, S., Venema, Y. and Goldblatt, R. and Grice, M, and some new duality theorems for the category of contact algebras and for the category of complete contact algebras.
Some Generalizations of Fedorchuk Duality Theorem -- I
推广V. V. Fedorchuk的对偶定理，证明了以下四类情形的Stone型对偶定理：它们都以局部紧的T2空间为对象，它们的态射分别是连续骨架映射、拟开完美映射、开映射、开完美映射.特别地，证明了所有紧T2空间及其之间的所有开映射范畴的Stone型对偶定理.我们还得到了这四个范畴的等价定理。这些定理的版本的满子范畴的这些范畴作为对象的所有局部紧连通T2空间制定以及。
A De Vries-type Duality Theorem for Locally Compact Spaces -- II
本文给出了第一部分的方法和结果以及M。Stone，H. de Vries，P. Roeper给出。特别是：得到了Stone对偶定理的一些推广，证明了局部切触布尔代数的一个完备定理，给出了Birkhoff问题72的Ponomarev解的一个直接证明，并描述了与（零维）Eberlein紧空间是余绝对的空间.
We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for locally small spaces with bounded continuous mappings.
We extend Exel's ample tight groupoid construction to general locally compact étale groupoids in the Hausdorff case. Moreover, we show how inverse semigroups are represented in this way as 'pseudobases' of open bisections, thus yielding a duality which encompasses various extensions of the classic Stone duality.
Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category $\bf KHaus$ of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category $\bf Tych$ of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories,…
We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices are those omitting certain types. (3) Compact clopen pseudobasic posets are characterized by separativity. We also show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations.
By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras. In our recent article, we have extended de Vries duality to completely regular spaces by generalizing de Vries algebras to de Vries extensions. To illustrate the utility of this point of view, we show how to use this new duality to obtain algebraic counterparts of normal and locally compact Hausdorff spaces in the form of de Vries extensions that are subject to additional axioms which encode the desired topological property. This, in particular, yields a different perspectiv…
As it was shown in the first part of this paper, there exists a duality between the category DSkeLC (introduced there) and the category SkeLC of locally compact Hausdorff spaces and continuous skeletal maps. We describe here the subcategories of the category DSkeLC which are dually equivalent to the following eight categories: all of them have as objects the locally compact Hausdorff spaces and their morphisms are, respectively, the injective (respectively, surjective) continuous skeletal maps, the injective (resp., surjective) open maps, the injective (resp., surjective) skeletal perfect maps…
For a compactification $αX$ of a Tychonoff space $X$, the algebra of all functions $f\in C(X)$ that are continuously extendable over $% αX$ is denoted by $C_α(X)$. It is shown that, in a model of $\textbf{ZF}$, it may happen that a discrete space $X$ can have non-equivalent Hausdorff compactifications $αX$ and $γX$ such that $% C_α(X)=C_γ(X)$. Amorphous sets are applied to a proof that Glicksberg's theorem that $βX\times βY$ is the Cech-Stone compactification of $X\times Y$ when $X\times Y$ is a Tychonoff pseudocompact space is false in some models of $\mathbf{ZF}$. It is noticed that if all T…
It is a classic result in modal logic that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality. Our goal is to further generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main n…
We show that the locally strongly sober spaces are exactly the coherent sober spaces that are weakly Hausdorff in the sense of Keimel and Lawson. This allows us to describe their Stone duals explicitly. As another application, we show that weak Hausdorffness is a sufficient condition for lenses and of quasi-lenses to form homeomorphic spaces, generalizing previously known results.
We unify several extensions of the classic Stone duality due to Grätzer, Hoffman-Lawson and Jung-Sünderhauf. Specifically we show that U-bases of locally compact sober spaces are dual to <-distributive v-predomains, where < is a transitive relation representing compact containment.
Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal contact algebras with suprema-preserving Boolean homomorphisms which reflect the contact relation.
The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic $\vDash_Δ$ associated with an infinitary variety $Δ$ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of $\vDash_Δ$, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic $\vdash_Δ$ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence associated with $\vdash_Δ$ c…
De Vries duality yields a dual equivalence between the category of compact Hausdorff spaces and a category of complete Boolean algebras with a proximity relation on them, known as de Vries algebras. We extend de Vries duality to completely regular spaces by replacing the category of de Vries algebras with certain extensions of de Vries algebras. This is done by first formulating a duality between compactifications and de Vries extensions, and then specializing to the extensions that correspond to Stone-Čech compactifications.
紧T2空间X具有复Stone-Weierstrass性质（CSWP）当且仅当它满足复形式的Stone-Weierstrass定理. W. Rudin证明了所有的离散空间都有CSWP。我们描述了一些技术证明某些非分散空间有CSWP。特别地，如果X是紧序空间和紧散布空间的乘积，则X有CSWP当且仅当X不包含康托尔集的副本。
In this article, we continue our study of Baire one functions on a topological space $X$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, it is observed that $X$ may not be embedded inside this structure space. This observation inspired us to build a space $\mathcal{M}(B_1(X))/\sim$, from the structure space of $B_1(X)$ and to show that $X$ is densely embedded in $\mathcal{M}(B_1(X))/\sim$. It is further established that it is a $T_0$-compactification of $X$. Such compactification of $X$ possesses the ex…
A Whiteheadian-type description of Euclidean spaces, spheres, tori and Tychonoff cubes
20世纪初，A. N. Whitehead和T.拉古纳提出了一种新的空间理论，称为{ \em region-based theory of space}。他们并没有以详细的数学形式提出自己的想法。1997年，P. Roeper证明了局部紧的T2空间是双射对应的（直到同胚和同构）正确表示Whitehead的{ \em区域}和{ \em接触关系}的代数对象，推广了de Vries关于紧T2空间的一个类似结果（注意，甚至紧T2空间和连续映射范畴的对偶也是由de Vries构造的）。最近，G。Dimov（它扩展了德弗里斯的对偶性上述）。本文利用de Vries和Dimov得到的对偶，直接（即不借助相应的拓扑空间）构造了欧几里得空间、球面、环面和Tychonoff立方体的对偶对象，这些代数对象完全刻画了上述拓扑空间.从而得到怀特海和德拉古纳关于欧几里得空间的原始哲学思想的数学实现。
We give a new proof of the Katětov-Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces. The key ingredient is how the ring of bounded continuous real-valued functions embeds in the ring of all bounded real-valued functions. In the compact case this embedding can be described by an appropriate statement, which we prove implies both the Katětov-Tong theorem and a version of the Stone-Weierstrass theorem. We then extend the Katětov-Tong theorem to all normal spaces by showing how to extend upper and lower semicontinuous re…
It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone-Cech compactification of $L^*\setminus\{0\}$, where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.
