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preprint2022arXiv

Totally disconnected semigroup compactifications of topological groups

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 obtained independently of Gelfand theory and well-known properties of the (typically non-totally disconnected) universal compactifications $G^{LUC}$, $G^{WAP}$ and $G^{AP}$, though we do employ Gelfand theory to clarify the relationship between these familiar universal compactifications and their totally disconnected counterparts.

preprint2022arXiv

Simultaneous Extension of Continuous and Uniformly Continuous Functions

The first known continuous extension result was obtained by Lebesgue in 1907. In 1915, Tietze published his famous extension theorem generalising Lebesgue's result from the plane to general metric spaces. He constructed the extension by an explicit formula involving the distance function on the metric space. Thereafter, several authors contributed other explicit extension formulas. In the present paper, we show that all these extension constructions also preserve uniform continuity, which answers a question posed by St. Watson. In fact, such constructions are simultaneous for special bounded functions. Based on this, we also refine a result of Dugundji by constructing various continuous (nonlinear) extension operators which preserve uniform continuity as well.

preprint2024arXiv

Twisted products: Enveloping actions and equivariant absolute neighborhood extensors

The classical notion of twisted product is studied in the context of partial actions, in particular, we show that the globalization of a partial action is a twisted product. In addition, we establish conditions for the metrizability of twisted products, and some homotopy and categorical properties are proved. Furthermore, sufficient conditions for the enveloping space to be an equivariant absolute neighborhood extensor are also studied.

preprint2024arXiv

Stone duality between condensed mathematics and algebraic geometry

We extend Stone duality to a fully faithful embedding of condensed sets into fpqc sheaves over an arbitrary field, which preserves colimits and finite limits. We study how familiar notions from condensed mathematics/topology and algebraic geometry correspond to each other under this form of Stone duality.

preprint2023arXiv

New bounds on the cardinality of Hausdorff spaces and regular spaces

Using weaker versions of the cardinal function $ψ_c(X)$, we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve $ψ_c(X)$ nor its variants at all. For example, we show if $X$ is regular then $|X|\leq 2^{c(X)^{πχ(X)}}$ and $|X|\leq 2^{c(X)πχ(X)^{ot(X)}}$, where the cardinal function $ot(X)$, introduced by Tkachenko, has the property $ot(X)\leq\min\{t(X),c(X)\}$. It follows from the latter that a regular space with cellularity at most $\mathfrak{c}$ and countable $π$-character has cardinality at most $2^\mathfrak{c}$. For a Hausdorff space $X$ we show $|X|\leq 2^{d(X)^{πχ(X)}}$, $|X|\leq d(X)^{πχ(X)^{ot(X)}}$, and $|X|\leq 2^{πw(X)^{dot(X)}}$, where $dot(X)\leq\min\{ot(X),πχ(X)\}$. None of these bounds involve $ψ_c(X)$ or $ψ(X)$. By introducing the cardinal functions $wψ_c(X)$ and $dψ_c(X)$ with the property $wψ_c(X)dψ_c(X)\leqψ_c(X)$ for a Hausdorff space $X$, we show $|X|\leqπχ(X)^{c(X)wψ_c(X)}$ if $X$ is regular and $|X|\leqπχ(X)^{c(X)dψ_c(X)wψ_c(X)}$ if $X$ is Hausdorff. This improves results of Sapirovskii and Sun. It is also shown that if $X$ is Hausdorff then $|X|\leq 2^{d(X)wψ_c(X)}$, which appears to be new even in the case where $wψ_c(X)$ is replaced with $ψ_c(X)$. Compact examples show that $ψ(X)$ cannot be replaced with $dψ_c(X)wψ_c(X)$ in the bound $2^{ψ(X)}$ for the cardinality of a compact Hausdorff space $X$. Likewise, $ψ(X)$ cannot be replaced with $dψ_c(X)wψ_c(X)$ in the Arhangel'skii-Sapirovskii bound $2^{L(X)t(X)ψ(X)}$ for the cardinality of a Hausdorff space $X$. Finally, we make several observations concerning homogeneous spaces in this connection.

preprint2022arXiv

Complete topologized posets and semilattices

In this paper we discuss the notion of completeness of topologized posets and survey some recent results on closedness properties of complete topologized semilattices.

preprint2011arXiv

Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak m$ satisfying ${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$ for all ${\mathfrak n}<{\mathfrak m}$, there exists a unique $\mathfrak m$-homogeneous universal metric space $\Ur_{\mathfrak m}$ of weight $\mathfrak m$. This object generalizes the classical Urysohn universal metric space $\Ur = \Ur_{\aleph_0}$. We show that for $\mathfrak m$ uncountable, the isometry group $\Iso(\Urm)$ with the topology of simple convergence is not a universal group of weight $\mathfrak m$: for instance, it does not contain $\Iso(\Ur)$ as a topological subgroup. More generally, every topological subgroup of $\Iso(\Urm)$ having density $<{\mathfrak m}$ and possessing the bounded orbit property $(OB)$ is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskij's 1990 result about the group $\Iso(\Ur)$ being a universal Polish group.

preprint2024arXiv

On the companion of spaces having dense, relatively countable compact subspaces

The notion of "pseudocompactness" was introduced by Hewitt. The concept of relatively countably compact subspaces were explored by Marjanovic to show that a $Ψ$-space is pseudocompact. A topological space is said to be DRC (DRS) iff it possesses a dense, relatively countably compact (or relatively sequentially compact, respectively) subspace. The concept of selectively pseudocompact game Sp(X) and the selectively sequentially pseudocompact game Ssp(X) were introduced by Dorantes-Aldama and Shakhmatov. They explored the relationship between the existence of a winning strategy and a stationary winning strategy for player P in these games. In particular, they observed that there exists a stationary winning strategy in the game Sp(X) (Ssp(X)) for Player P iff $X$ is DRC (or DRS, respectively). In this paper we introduce natural weakening of the properties DRC and DRS: a space $X$ is DRCo ( DRSo) iff there is a sequence $(D_n:n \in { ω})$ of dense subsets of $X$ such that every sequence $(d_n:n \in { ω} )$ with $d_n \in D_n$ has an accumulation point (or contains a convergent subsequence, respectively). These properties are also equivalent to the existence of some limited knowledge winning strategy on the corresponding games $Sp(X)$ and $Ssp(X)$. Clearly, DRS implies DRC and DRSo, DRC or DRSo imply DRCo. The main part of this paper is devoted to prove that apart from these trivial implications, consistently there are no other implications between these properties.

preprint2011arXiv

Ordinal Compactness

We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities of families. The most general form of this compactness notion depends on two ordinal parameters. In the particular case when the parameters are cardinal numbers, we get back a classical notion. Generalized to ordinal numbers, this notion turns out to behave in a much more varied way. We present many examples of spaces satisfying the very same cardinal compactness properties, but with a broad range of distinct behaviors, with respect to ordinal compactness. A much more refined theory is obtained for $T_1$ spaces, in comparison with arbitrary topological spaces. The notion of ordinal compactness becomes partly trivial for spaces of small cardinality.

preprint2022arXiv

Discrete Ultrafilters and Homogeneity of Product Spaces

An ultrafilter $p$ on $ω$ is said to be discrete if, given any function $f\colon ω\to X$ to any completely regular Hausdorff space, there is an $A \in p$ such that $f(A)$ is discrete. Basic properties of discrete ultrafilters are studied. Three intermediate classes of spaces $\mathscr R_1 \subset \mathscr R_2 \subset \mathscr R_3$ between the class of $F$-spaces and the class of van~Douwen's $βω$-spaces are introduced. It is proved that no product of infinite compact $\mathscr R_2$-spaces is homogeneous; moreover, under the assumption $\mathfrak d =\mathfrak c$, no product of $βω$-spaces is homogeneous.

preprint2022arXiv

Generation of adaptive cellular structures to surfaces with complex geometries using Homotopy functions and conformal transformation

The optimal use of resources has motivated the engineering community to employ controlled distribution of material within their structural designs, often relying on cellular and lattice porous structures. In this research work, a computational method is implemented to generate porous lattice structures conforming to surfaces with complex geometry. This method allows solving the problem of integrating cellular structures in engineering applications with complex macro-geometries. Examples of surface transformation are shown in this report. In addition, an applied model is shown where the effects of the transformation on the mechanical properties are analyzed.

preprint2022arXiv

Circular orderability and quandles

In this paper, we introduce the notion of circular orderability for quandles. We show that the set all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right (respectively left) orderings of a quandle embeds in its space of right (respectively left) circular orderings. Examples of quandles that are not left circularly orderable and examples of quandles that are neither left nor right circularly orderable are given.

preprint2022arXiv

Mildly version of Hurewicz Basis covering property and Hurewicz measure zero spaces

In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Kočinac, and Scheepers. A space $X$ is said to have mildly-Hurewicz property if for each sequence $\langle \mathcal{U}_n : n\in ω\rangle$ of clopen covers of $X$ there is a sequence $\langle \mathcal{V}_n : n\in ω\rangle$ such that for each $n$, $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and for each $x\in X$, $x$ belongs to $\bigcup\mathcal{V}_n$ for all but finitely many $n$. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces.

preprint2024arXiv

Connected properties of the sublinear Higson corona of $\Bbb R^n$

In this paper, for the $n$-dimensional Euclidean space $\Bbb R^n$ with the usual metric and $n\geq2$, we show that the sublinear Higson corona of $\Bbb R^n$ is not locally connected at any point and is mutually aposyndetic.

preprint2024arXiv

Products of Directed Sets with Calibre $(ω_1, ω)$

A directed set $P$ is calibre $(ω_1, ω)$ if every uncountable subset of $P$ contains an infinite bounded subset. $P$ is productively calibre $(ω_1, ω)$ if $P \times Q$ is calibre $(ω_1, ω)$ for every directed set $Q$ with calibre $(ω_1, ω)$, and $P$ is powerfully calibre $(ω_1, ω)$ if the countable power of $P$ is calibre $(ω_1, ω)$. It is shown that (1) uncountable products are calibre $(ω_1, ω)$ only in highly restrictive circumstances, (2) many but not all $\sum$-products of calibre $(ω_1, ω)$ directed sets are calibre $(ω_1, ω)$, (3) there are directed sets which are calibre $(ω_1, ω)$ but neither productively nor powerfully calibre $(ω_1, ω)$, and (4) there are directed sets which are powerfully but not productively calibre $(ω_1, ω)$. As an application, the position is established of $\sum ω^{ω_1}$ in the Tukey order among Isbell's classical 10 directed sets.

preprint2024arXiv

When ideals properly extend the class of Arbault sets

In this article we continue the investigation of generalized version of Arbault sets, that was initiated in \cite{DGT} but look at the picture from the most general point of view where ideals come into play. While Arbault sets can be naturally associated with the Frechet ideal $Fin$, in \cite{DGT} it was observed that when $Fin$ is replaced by the natural density ideal $\mathcal{I}_d$ one can obtain a strictly larger class of trigonometric thin sets containing Arbault sets. From the set theoretic point of view a natural question arises as whether one can broaden the picture and specify a class of ideals (instead of a single ideal) each of which would have the similar effect. As a natural candidate, we focus on a special class of ideals, namely, non-$snt$ ideals ($snt$ stands for ``strongly non translation invariant") which properly contains the class of translation invariant ideals ($\varsupsetneq Fin$) and happens to contain ideals generated by simple density functions as also certain non-negative regular summability matrices (but not all) which can be seen from \cite{DG6}. We consider the resulting class of $\mathcal{I}$-Arbault sets and it is observed that for each such ideal, the class of $\mathcal{I}$-Arbault sets not only properly contains the class of classical Arbault sets \cite{Ar} but also a large subfamily of $\mathbf{N}$-sets (also called ``sets of absolute convergence") \cite{Ft} while being contained in the class of weak Dirichlet sets. %In particular it properly contains the family of $\mathbf{N}_0$-sets which have been extensively used in the literature (see \cite{Ar, Ka, Ko}). Though distinct from the class of $\mathbf{N}$-sets, this happens to be a new class strictly lying between the class of Arbault sets and the class of weak Dirichlet sets.

preprint2024arXiv

Local compactness in MT-algebras

In our previous work, we introduced McKinsey-Tarski algebras (MT-algebras for short) as an alternative pointfree approach to topology. Here we study local compactness in MT-algebras. We establish the Hofmann-Mislove theorem for sober MT-algebras, using which we develop the MT-algebra versions of such well-known dualities in pointfree topology as Hofmann-Lawson, Isbell, and Stone dualities. This yields a new perspective on these classic results.

preprint2015arXiv

Some Properties of Mappings on Generalized Topological Spaces

This paper considers generalizations of open mappings, closed mappings, pseudo-open mappings, and quotient mappings from topological spaces to generalized topological spaces. Characterizations of these classes of mappings are obtained and some relationships among these classes are established.

preprint2024arXiv

Hofmann-Mislove through the Lenses of Priestley

We use Priestley duality to give a new proof of the Hofmann-Mislove Theorem.

preprint2023arXiv

Ellis groups in model theory and strongly generic sets

Assume $G$ is a group and $\mathcal{A}$ is an algebra of subsets of $G$ closed under left translation. We study various ways to understand the Ellis group of the $G$-flow $S(\mathcal{A})$ (the Stone space of $\mathcal{A}$), with particular interest in the model-theoretic setting where $G$ is definable in a first order structure $M$ and $\mathcal{A}$ consists of externally definable subsets of $G$. In one part of the thesis we explore strongly generic sets. Maximal algebras of such sets are shown to carry enough information to retrieve the Ellis group. A subset of $G$ is strongly generic if each non-empty Boolean combination of its translates is generic. Trivial examples include what we call *periodic* sets, which are unions of cosets of finite index subgroups of $G$. We give several characterizations of strongly generic sets, in particular, we relate them to almost periodic points of the flow $2^G$. For groups without a smallest finite index subgroup we show how to construct non-periodic strongly generic subsets in a systematic way. When $G$ is definable in a model $M$, a definable, strongly generic subset of $G$ will remain as such in any elementary extension of $M$ only if it is strongly generic in $G$ in an adequately uniform way. Sets satisfying this condition are called *uniformly strongly generic*. We analyse a few examples of these sets in different groups. In the second part we assume that $G$ is a topological group and consider a particular algebra of its subsets denoted $\mathcal{SBP}$. It consists of subsets of $G$ that have the *strong Baire property*, meaning nowhere dense boundary. We explicitly describe the Ellis group of $S(\mathcal{A})$ for an arbitrary subalgebra $\mathcal{A}$ of $\mathcal{SBP}$ under varying assumptions on the group $G$, including the case when $G$ is a compact topological group. [...] (Full abstract in the article)

preprint2023arXiv

Homogeneous locally compact spaces

This is a survey of the recent results and unsolved problems about locally compact homogeneous metric spaces. Mostly, homogeneous finite-dimensional $ANR$-spaces are discussed.

preprint2022arXiv

Locally closed sets and submaximal spaces

A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $βX$, the Stone-Čech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $βX=X$. We also prove that if $\upsilon X$, the Hewitt realcompactification of $X$, is submaximal and first countable and $X$ is without isolated point, then $X$ is realcompact and hence $\upsilon X=X$. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if $\upsilon X$ is a submaximal space, then $X$ is a pseudo-finite $μ$-compact space. An example is given which shows that $X$ may be submaximal but $\upsilon X$ may not be submaximal. Given a topological space $(X,{\mathcal T})$, the collection of all locally closed subsets of $X$ forms a base for a topology on $X$ which is denotes by ${\mathcal T_l}$. We study some topological properties between $(X,{\mathcal T})$ and $(X,{\mathcal T_l})$, such as we show that a) $(X,{\mathcal T_l})$ is discrete if and only if $(X,{\mathcal T})$ is a $T_D$-space; b) $(X,{\mathcal T})$ is a locally indiscrete space if and only if ${\mathcal T}={\mathcal T_l}$; c) $(X,{\mathcal T})$ is indiscrete space if and only if $(X,{\mathcal T_l})$ is connected. We see that, in locally indiscrete spaces, the concepts of $T_0$, $T_D$, $T_\frac{1}{2}$, $T_1$, submaximal and discrete coincide. Finally, we prove that every clopen subspace of an lc-compact space is lc-compact.

preprint2022arXiv

Spaces of countable free set number and PFA

The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F(X) = ω$ we have $|X| \le w(X)^ω$; in particular, $w(X) \le \mathfrak{c}$ implies $|X| \le \mathfrak{c}$. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. In fact, this space $X$ has the {\em strongly anti-Urysohn} (SAU) property that any two infinite closed sets in $X$ intersect, which is much stronger than $F(X) = ω$. Moreover, any non-empty open set in $X$ also has size $2^\mathfrak{c}$, and thus answers one of the main problems of \cite{JShSSz} by providing in ZFC a SAU space with no isolated points.

preprint2023arXiv

On weak $I^K$-Cauchy sequences in normed spaces

In this paper, we study on weak $I^K$-Cauchy condition as a generalization of weak $I^*$-Cauchy condition in a normed space. We investigate the relationship between weak $I$-Cauchy and weak $I^K$-Cauchy sequences using $AP(I,K)$-condition. Also we study on weak* $I^K$-Cauchy condition and weak $I^K$-divergence of sequences in the same space.