Kenneth Kunen
C(X) denotes the space of continuous complex-valued functions on the compact Hausdorff space X. X has the CSWP if every subalgebra of C(X) which separates points and contains the constant functions is dense in C(X). W. Rudin showed that all scattered X have the CSWP. We describe a class of non-scattered X with the CSWP; by another result of Rudin, such X cannot be metrizable.
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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…
A Stone-Weierstrass approximation theorem for monotone functions
本文提出一个关于单调函数的逼近定理，并证明正非递减有理函数可以一致逼近所有连续非递减函数。
The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.
We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results about those compact Hausdorff spaces which have scattered-to-one maps onto compact metric spaces.
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.
We investigate the question of when a topological space $X$ has the $\textit{Generalized Bolzano-Weierstrass property}$: every sequence of subsets of $X$ has a convergent subsequence (in the sense of Kuratowski).
Assuming Jensen's principle diamond, there is a compact Hausdorff space X which is hereditarily Lindelof, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The diamond example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot contain the two earlier examples of failure of the CSWP, which were totally disconnected -- specifically, the Cantor set (W. Rudin) and beta N (Hoffman and Singer).
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…
A Boolean algebra $\mathcal A$ has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on $\mathcal A$ is uniformly bounded. Assuming the Diamond Principle $\Diamond$, we will construct an example of a minimally generated Boolean algebra $\mathcal A$ with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true - again under $\Diamond$ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The r…
We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Ban…
El mar que envuelve a las piedras: espacios de Stone, profinitos y su papel en la aritmética contemporánea
本文探讨Stone空间与profinite空间在算术几何中的核心作用，通过布尔代数与Stone空间的构造，阐述其在数论和Condensed Mathematics中的应用。
Some remarks on the projective properties of Menger and Hurewicz
本文针对Menger与Hurewicz性质的投射版本，给出类似于经典情形中借助Čech-Stone紧化$βX$描述Tychonoff空间放置方式的刻画。
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…
We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is is equivalent to the underlying topological space being a Stone space. Condensing the core idea of those classical results, we introduce the concept of affine boundedness for an arbitrary unive…
On points avoiding measures
该研究定义了拓扑空间中“避免测度的点”的概念，指出其否定是支撑严格正测度性质的局部版本，并在一般拓扑及Stone-Čech紧化剩余空间$ω^\ast$场景下开展相关研究。
We show that for a normal locally-${\mathscr P}$ space $X$ (where ${\mathscr P}$ is a topological property subject to some mild requirements) the subset $C_{\mathscr P}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with ${\mathscr P}$, is a subalgebra of $C_b(X)$ isometrically isomorphic to $C_c(Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification $βX$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathscr P}(X)$ coinci…
Linear homeomorphisms of function spaces and the position of a space in its compactification
本文针对 Arhangel'skii 关于 Tychonoff 空间 Menger 性质是否被函数空间同胚保持的问题，在线性同胚情形下给出肯定回答，并提出通过分析空间在其紧致化中的位置来研究函数空间线性同胚不变量的方法。
Let $X$ and $Y$ be pseudocompact spaces and let the function $Φ: X\times Y\to \mathbb R$ be separately continuous. The following conditions are equivalent: (1) there is a dense $G_δ$ subset of $D\subset Y$ so that $Φ$ is continuous at every point of $X\times D$ (Namioka property); (2) $Φ$ is quasicontinuous; (3) $Φ$ extends to a separately continuous function on $βX\times βY$. This theorem makes it possible to combine studies of the Namioka property and generalizations of the Eberlein-Grothendieck theorem on the precompactness of subsets of function spaces. We also obtain a characterization of…
Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these continuous mappings -- also tools from Model Theory. We illustrate by showing that the Čech-Stone remainder $[0,\infty)$ has a universality property akin to that of $N^*$; a theorem of Maćkowiak and Tymchatyn implies it own generalization to non-metric continua; and certain concrete compact spaces need not be continuous images of $N^*$.
Remarks on SHD spaces and more divergence properties
本文回答了关于 SHD 空间的若干问题，构造了非 SHD 空间但其超空间与紧化为 SHD 的实例，并提出了 WSHD 和 OHD 等新变体性质以探讨空间与其超空间的关系。
The Nikodym property and filters on $ω$
本文研究ω上滤子对应的拓扑空间与布尔代数Nikodym性质的关联，刻画了AN类理想的等价条件，分析了其Katětov序结构及密度理想的相关等价性质。
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