Alex Chigogidze
We show that for each countable simplicial complex P the following conditions are equivalent: (1) $P \in AE(X)$ iff $P \in AE(βX)$ for any space X; (2) There exists a P-invertible map of a metrizable compactum X with $P \in AE(X)$ onto the Hilbert cube.
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Generalizing de Vries Compactification Theorem and strengthening Leader Local Compactification Theorem, we describe the partially ordered set $(\LL(X),\le)$ of all (up to equivalence) locally compact Hausdorff extensions of a Tychonoff space $X$. Using this description, we find the necessary and sufficient conditions which has to satisfy a map between two Tychonoff spaces in order to have some kind of extension over arbitrary given in advance Hausdorff local compactifications of these spaces; we regard the following kinds of extensions: open, quasi-open, skeletal, perfect, injective, surjectiv…
Generalizing a theorem of Ph. Dwinger, we describe the partially ordered set of all (up to equivalence) zero-dimensional locally compact Hausdorff extensions of a zero-dimensional Hausdorff space. Using this description, we find the necessary and sufficient conditions which has to satisfy a map between two zero-dimensional Hausdorff spaces in order to have some kind of extension over arbitrary given in advance Hausdorff zero-dimensional local compactifications of these spaces; we regard the following kinds of extensions: continuous, open, quasi-open, skeletal, perfect, injective, surjective. I…
Let $X$ be a space. A space $Y$ is called an extension of $X$ if $Y$ contains $X$ as a dense subspace. For an extension $Y$ of $X$ the subspace $Y\backslash X$ of $Y$ is called the remainder of $Y$. Two extensions of $X$ are said to be equivalent if there is a homeomorphism between them which fixes $X$ pointwise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous mapping of $Y'$ into $Y$ which fixes $X$ pointwise. Let $P$ be a topological property. An extension $Y$ of $X$ is called a $P$-extension of $X$ if it has $P$. If $P$ is compactness …
We construct a locally finite connected graph whose Freudenthal compactification is universal for the class of completely regular continua, a class also known in the literature under the name thin or graph-like continua.
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.
After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Cech compactification βN, we establish some results about one of such extensions. This provides us with one possible divisibility relation on βN and we introduce a few more, defined in a natural way. For some of them we find equivalent conditions for divisibility. Finally, we mention a few facts about prime and irreducible elements of (βN, \cdot). The motivation behind all this is to try to translate problems in number theory into βN.
We introduce the notion of an introverted Boolean algebra $\cal B$ of closed-and-open subsets of a topological group $G$, show that the associated Stone space $(ν_{\cal B} G, ν_{\cal B})$ is a totally disconnected semigroup compactification of $G$, and show that every totally disconnected semigroup compactification of $G$ takes this form. We identify and study the universal totally disconnected semigroup compactification, the universal totally disconnected semitopological semigroup compactification and the universal totally disconnected group compactification of $G$. Our main results are obtai…
Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions
本文刻画了乘积空间上可连续延拓到其紧化的连续函数，并由此将任意拓扑结构的Bohr紧化实现为殆周期函数所成交换Banach代数的Gelfand空间，同时应用于殆周期函数空间之间等距映射的表示。
A space Y is called an extension of a space X if Y contains X as a dense subspace. An extension Y of X is called a one-point extension if Y-X is a singleton. Compact extensions are called compactifications and connected extensions are called connectifications. It is well known that every locally compact non-compact space has a one-point compactification (known as the Alexandroff compactification) obtained by adding a point at infinity. A locally connected disconnected space, however, may fail to have a one-point connectification. It is indeed a long standing question of Alexandroff to characte…
In contexts in which the principle of dependent choice may not be available, as toposes or Constructive Set Theory, standard locale theoretic results related to complete regularity may fail to hold. To resolve this difficulty, B. Banaschewski and A. Pultr introduced strongly regular locales. Unfortunately, Banaschewski and Pultr's notion relies on non-constructive set existence principles that hinder its use in Constructive Set Theory. In this article, a fully constructive formulation of strong regularity for locales is introduced by replacing non-constructive set existence with coinductive se…
This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of $X$ which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity. Based on this foundational wor…
Nagata conjectured that every $M$-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset $P$ of size $2^ω$ such that in $V^P$ Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space $X\in V$ is an $M$-space in $V^P$ then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in $V^P$). In fact, w…
A generalized topology in a set $X$ is a collection $\text{Cov}_X$ of families of subsets of $X$ such that the triple $(X,\bigcup \text{Cov}_X,\text{Cov}_X)$ is a generalized topological space in the sense of Delfs and Knebusch. In this work, notions of topological and admissible compactness of generalized topologies are introduced to begin and investigate a theory of compactifications, in particular, of Wallman type in the category of weakly normal generalized topological spaces. Among other facts, we prove in ZF that the ultrafilter theorem (in abbreviation UFT) holds if and only if all Wall…
In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann. Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained of the locales of which the Stone-Cech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of dependent choice. In this paper I show that this characterization continues to hold over the standard system CZF plus REA, thus removing in particular any depend…
\begin{abstrac} Let $(X,T) $ be a topological space, and $^{*}X$ a non--standard extension of $X$. There is a natural ``standard'' topology $^{S}T$ on $^{*}X$ generated by $^{*}G$, where $G\in T$. The topological space $(^{*}X,^{S}T) $ will be used to study compactifications of $(X,T)$ in a systematic way.
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…
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-dimensio…
We verified that the existence of a maximal ideal of height 0 in a p-adic algebra in a certain class is independent of the axiom of ZFC. We established the theory on a P-point in the boundary of a topological space in the universal totally disconnected Hausdorff compactification. It is quite similar with the theory on a P-point in the boundary of a topological space in the Stone-Cech compactification. The latter theory relies on the real analysis, and the reason why the real analysis works for it is because the Stone-Cech compactification has the lifting property for a real bounded continuous …
Extensions and factorizations of topological and semitopological universal algebras
本文研究拓扑与半拓扑代数运算向Stone-Čech紧化的扩张及连续函数经同态到可度量化代数的分解，重点关注伪紧与紧代数。
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…
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