Papers

In this paper, we introduce two new classes of cyclic-type mappings, namely, co-cyclic mappings and sub-cyclic mappings, defined on the union of two subsets of a metric space. Under suitable Jungck-type contractive conditions and compatibility assumptions, we establish unique common fixed point theorems for these mappings in complete metric spaces. The proofs are based on the construction of appropriate sequences, the associated contractive conditions, and the completeness of the underlying metric space. Examples are provided to illustrate the proposed concepts and to demonstrate the applicability of the main results. Finally, some open questions and possible directions for further research concerning co-cyclic and sub-cyclic mappings are presented.

We give a negative answer to the question, posed by Erné, whether every $3$-distributive lattice is $\omega$-distributive. More precisely, we exhibit a continuous frame that is $\kappa$-distributive for every integer $\kappa\geq 2$, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based $T_0$ topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of $\omega$-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Erné's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

We investigate the finite ultrametric spaces $(X,d)$ that have a given cardinality of the center of distances and a minimal cardinality of the set $X$. It is shown that such spaces are isometric if and only if their centers of distances are the same. The representing trees of these spaces are characterized up to isomorphism.

Perhaps surprisingly, there are infinite types that can be exhaustively searched mechanically in finite time. We use ideas from topology to build plenty of them, referring to searchable types as compact types, and we use ordinals to measure their logical complexity. We consider two systems of ordinal notations under which a single notation denotes both a discrete ordinal and a compact one, with an embedding of the former into the latter whose image has empty complement. A boolean valued function decides which points in the image of the embedding are isolated and which are topological limit points. The first system consists of the traditional Brouwer codes and the second is an inductive-recursive universe generalizing them. The discrete ordinals so obtained are trichotomous, and the compact ones have the least element property for complemented subsets, but these two desirable properties cannot be fulfilled simultaneously in a constructive setting. The ordinals obtained from Brouwer codes further enjoy a boolean Leibniz principle, which has the notion of total separatedness as its topological counterpart. This extends previous work from Gödel's system T to intensional Martin-Löf type theory with univalent universes, and is formalized in Agda in the TypeTopology repository.

The first aim of this article is to extend the notion of the Gromov product to the class of partial metric spaces. A relation between $\delta$-hyperbolicity of a partial metric space and $\delta$-hyperbolicity of metric spaces endowed with the metrics defined via that partial metric is also established. The second one is to analyze the Gromov product in the wide class of metric spaces in which metrics are defined via Chebyshev sets. Midpoint mappings in this class of spaces will be also examined.

We prove that the space of isometry classes of nonempty compact metric spaces, equipped with the Gromov–Hausdorff distance, is homeomorphic to the real separable infinite-dimensional Hilbert space. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite-dimensional local approximations that control all pairwise distances. These approximations yield the absolute retract property for all metrizable spaces. We also prove that any countable family of continuous maps from compact metrizable spaces can be approximated, with respect to a prescribed open cover, by maps whose images form a discrete family.

Settling a central problem in set-theoretic topology, and many related problems, we prove that there is, consistently, a regular Lindelöf space of cardinality $\aleph_1$ that is not a D-space. The existence of such a space is independent of ZFC; we establish its equivalence to a weak form of the Continuum Hypothesis. The proof of the main theorem is adjustable to produce consistent examples with a variety of stronger properties.

We study the problem of universal approximation of continuous maps between topological vector spaces by neural networks. We provide a universal approximation theorem for maps between locally convex topological vector spaces with and without paracompactness assumptions. We extend this result to continuous maps between non-locally convex topological vector spaces on compact sets under the assumption of finite topological dimension.

In this short note, we prove that, for any pair of finite groups $G$ and $H$, there exist a finite $T_0$-space $X$ and a semiflow $\varphi\colon [0,\infty)\times X\to X$ such that $\operatorname{Aut}(X)\cong G$, whereas $\operatorname{Aut}(\varphi_t(X))\cong H$ for every $t>0$. Thus, the symmetry group of a finite $T_0$-space and those of all its positive-time images can be prescribed independently.

A topological group is functionally balanced if every bounded real-valued left uniformly continuous function is right uniformly continuous. We prove that every Hausdorff functionally balanced group has coinciding left and right uniformities, giving an affirmative answer to the Itzkowitz problem. Consequently, the classes of SIN, SFSIN and FSIN groups coincide.

We introduce the notion of perfect regularity as an appropriate weakening of perfect normality, both for spaces and locales. Various characterizations are given, using Dedekind-MacNeille completions, injective hulls, and sublocales. We place the new class of perfectly regular frames among various well-studied classes of frames. We also introduce the construction of perfect regularization of a completely regular frame, compare it to Isbell's well-known booleanization construction, and argue that it is at least as important as the latter.

Horn conjectured that whenever $K$ is a nonempty compact convex subset of a Banach space and $F,G:K\to K$ are continuous commuting maps, there exists $x\in K$ such that $F(x)=G(x)$. We give a counterexample in a separable Hilbert space. The construction uses an exactly commuting ladder of coincidence-free maps between finite trees, due to Oversteegen and Rogers.

The classical Banach principle relies on contractive sequences being Cauchy. Gregori and Sapena (2002) established a fuzzy analogue but required this as an extra hypothesis, leaving it as an open problem. Gregori et al. (2020) later asked whether strictly fuzzy $\psi$-contractive sequences are necessarily Cauchy. We settle both problems negatively via a single counterexample: in a constructed GV-fuzzy metric space, there exists a sequence which is both GS-contractive and strictly fuzzy $\psi$-contractive, yet not Cauchy. This shows that contractivity never implies Cauchyness in general GV-fuzzy metric spaces, and additional structural conditions in fuzzy fixed point theorems are indispensable.

The Poincaré polynomials of the Deligne-Mumford space $\overline{M_{0,n}}$ of stable genus 0 curves have been widely studied by several authors such as Keel and Manin. These polynomials can be computed via a recursive formula that is combinatorial in nature, and their exponential generating functions satisfy elegant functional and differential equations. In this paper, we state some combinatorial formulas to the Poincaré polynomials of Hassett's heavy-light moduli spaces $\overline{M_{0,w_{m,n}}}$, with $m$ heavy marked points and $n-m$ light marked points. We express the Poincaré polynomials recursively in terms of the Möbius function of a certain lattice of set partitions. In the case of $m=2$, we get a Losev-Manin space. We give an explicit formula for the Poincaré polynomial or $\overline{M_{0,w_{2,n}}}$ by counting ordered set partitions; and give a recursive formula for this polynomial similar to that in the setting of $\overline{M_{0,n}}$. We prove this recursive formula using geometric and topological properties of the stratification of Losev-Manin spaces; and use an exponential generating function to simplify this recursive formula. Finally, we give a remarkable generalization to the ordered Bell numbers and a recurrence relation for this generalization.

We study the relationship between quasi-uniform spaces, topological spaces, and quantale-valued metric spaces. Our main result is a metrization theorem establishing an equivalence between the category of quasi-uniform spaces and a category of quantale-valued metric spaces. We also obtain a quantale-based metrization theorem for arbitrary topological spaces that refines existing constructions. These results identify quasi-uniformities as the appropriate qualitative counterpart of quantale-valued metrics. Building on this correspondence, we show that the Hausdorff-Smyth monad on quantale-valued metric spaces, which is used in quantitative robustness analysis, arises as a lifting of a corresponding monad on quasi-uniform spaces along the equivalence. This provides a unified categorical framework connecting topology, quasi-uniformity, and quantitative robustness analysis.

In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree $T_{NU}(X)$ associated with each separable Banach space $X$ which does not contain an isomorphic copy of $C(2^\mathbb{N})$. Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of $C(2^\mathbb{N})$. Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if $H$ is any separable Hilbert space, then $o(T_{NU}(H)) = {\omega} + 1$ when taken with respect to the Haar basis for $C(2^\mathbb{N})$, demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.

We establish the cardinal inequality \(|X|\leq 2^{t(X)H\psi(X)}\) for every Hausdorff SDL space \(X\), where \(t(X)\) and \(H\psi(X)\) denote the tightness and the Hausdorff pseudocharacter of \(X\), respectively. Since both invariants are bounded by \(\chi(X)\), this yields \(|X|\leq 2^{\chi(X)}\). As a consequence, every first-countable Hausdorff strongly cellular–Lindelöf space has cardinality at most the continuum. These results answer Questions~2.1 and~2.2 of Bella and Spadaro. An intermediate result is a uniform bounded-decomposition property for SDL spaces; in particular, their strict quasi–Lindelöf number satisfies \(\sqL(X)\leq t(X)\).

We strengthen a recent selection game equivalence proved by L. Chiozini involving spaces and their associated spaces of continuous real-valued functions. This strengthening establishes that the Rothberger game is perfect- and Markov-information dual to the strong closed discrete game on the space of real-valued continuous functions. In the process, we extend the theory of strategic equivalences for the second player in variations of the Menger and Rothberger games involving large covers while noting explicitly how large covers present serious obstacles to applying strategy translation results used by the authors in similar scenarios in previous papers. We also briefly comment on subbasic selection games introduced by D. Guerrero Sánchez and V.V. Tkachuk.

We study the partition regular properties of topological groups, proving several extensions of Hindman theorem where monochromatic sets of finite sums are required to satisfy additional topological constraints. In particular, our results imply that for every nowhere dense set $C \subseteq \mathbb{R}^n$, there exists an open set $P \supseteq C$ such that for any finite coloring of $\mathbb{Q}^n \setminus P$, there is a family $\mathcal{A}$ of sequences in $\mathbb{Q}^n \setminus P$ which satisfies the following properties: (i) for each $A\in\mathcal A$, the set $\operatorname{FS}(A)$ of finite sums of $A$ is a closed discrete subset of $\mathbb R^n$; (ii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is monochromatic; and (iii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is dense in an open unbounded subset of $\mathbb R^n$.

In this paper it is proved that there exists (in $ZFC$) a separable pseudocompact weak $\lambda$-space $X$ which is not $\Delta_1$-space. It follows that there exists a space $X$ such that $B_1(X,[0, 1])$ is Choquet, while $B_1(X)$ is not Choquet. Also it is proved that there exists a $\gamma$-set $X$ which is not weak $\lambda$-set. It follows that there exists a zero-dimensional separable metrizable space $X$ such that $B_1(X)$ is Baire, while $B_1(X, [0,1])$ is not Choquet. These results answers the previously posed questions.

Let $\theta$ be an ordinal and let $(N_\alpha)_{\alpha\leq\theta}$ be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group $G$, with $N_0=\{e\}$. Write $\rho G$ for the Raikov completion of $G$, and put $K_\alpha=\overline{N_\alpha}^{\rho G}$ and $H_\alpha=GK_\alpha$. The successive and limit differences of the groups $H_\alpha$ mark the first stages at which points of $\rho G$ become old quotient points. For every nonzero ordinal $\theta$ and every set $S\subseteq(0,\theta]$, there is a Hausdorff pseudocompact Boolean group with a continuous tower of closed normal subgroups such that every member of the tower is pseudocompact and the nonempty strata occur exactly at the stages in $S$. The terminal quotient can simultaneously be chosen isomorphic to $\mathbb Z/2\mathbb Z$, and for every $\delta<\theta$ the successive factor $N_{\delta+1}/N_\delta$ is compact exactly when $\delta+1\notin S$. Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. For a closed normal subgroup $N$ of $G$, let $\widehat q:\rho G\to\rho(G/N)$ extend the quotient homomorphism and put $K=\overline N^{\rho G}$. Over $gN\in G/N$, the fiber in $\rho G\setminus G$ is $g(K\setminus N)$, homeomorphic to $\rho N\setminus N$, while for $y\in\widehat q(\rho G)\setminus(G/N)$ the entire $\widehat q$-fiber lies in $\rho G\setminus G$ and is homeomorphic to $\rho N$. For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. Further results concern pseudocompact Raikov remainders and include countable Abelian examples.

The topological Halmos–von Neumann theorem is a fundamental result of topological dynamics that completely classifies continuous actions of abelian topological groups which are ergodic and have discrete spectrum. It rests on a correspondence between such actions (up to the choice of a distinguished point), compactifications of the acting group and subgroups of its dual group. We generalize both the classical statement and the underlying categorical equivalences to the framework of open group bundles over locally compact base spaces. As a byproduct, we establish a Pontryagin-type duality between étale Hausdorff abelian group bundles and open proper Hausdorff abelian group bundles, extending the classical duality between discrete and compact abelian groups.

We show that the homeomorphism groups of the following spaces have precisely $2^{2^{\aleph_0}}$ maximal subgroups: the rational numbers $\mathbb{Q}$, the Baire space $\mathbb{N}^{\mathbb{N}}$, the space $\mathbb{N}\times 2^{\mathbb{N}}$ where $2^{\mathbb{N}}$ is the Cantor set, the ordinal $\omega^2$ under its order topology, and the Sorgenfrey line $\mathbb{S}$. More generally, we find sufficient conditions on a group $G$ acting on a topological space which imply that $G$ has at least $2^{2^{\aleph_0}}$ maximal subgroups. Moreover, if the groups $\operatorname{Homeo}(\mathbb{Q})$ and $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ are equipped with the pointwise topology, then it is shown that $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ has precisely $2^{\aleph_0}$ open maximal subgroups, and $\operatorname{Homeo}(\mathbb{Q})$ has precisely $\aleph_0$ open maximal subgroups and $2^{\aleph_0}$ closed maximal subgroups.

We study proper actions on homogeneous spaces of the group $\mathrm{B}(n)$ of invertible upper triangular matrices from a coarse-geometric viewpoint. We obtain sufficient conditions for the properness of the natural $L$-action on $\mathrm{B}(n)/H$, where $L,H$ are closed subgroups of $\mathrm{B}(n)$. We also give an explicit sufficient condition for properness of the $L$-action on a homogeneous space naturally identified with $\mathrm{B}(n)/H\cong \mathbb{R}^{n-1}$. In addition, we discuss extensions to the case where the ambient group $G$ is a subgroup of $\mathrm{B}(n)$ and $L,H$ are closed subgroups of $G$, with particular attention to the unipotent subgroup $\mathrm{N}(n)$.

In this paper, we address the dynamic construction problem for Malfatti circles with respect to a fixed triangle, using the LC and CC modules. We show that the configuration of the Malfatti circles constitutes a stable fixed point and provide lower bounds for the main parameters of the LC and CC modules. We now remark that all mathematical proofs in this paper were generated by ChatGPT.

This manuscript focuses on the topological behaviour of algebraic structures in $C_p(X)$. Closure of ideals in $C_p(X)$ is determined and are used to characterise various topological properties of the underlying Tychonoff space $X$. These include compact space, locally compact space, locally pseudocompact space, $P$-space, almost $P$-space and normal space. Moreover, it has been established that a space $X$ is totally separated space if and only if the set of all units in $C(X)$ is dense in $C_p(X)$. A subring of $C(X)$ is found to be dense in $C_p(X)$ when and only when it separates points.

We study the (canonical) 1st derivatives of proper submersions represented as height functions and belonging to a certain class: a proper map means a map the preimage of a compact set by which is always compact. We investigate their Reeb spaces. They are the quotient spaces defined by the equivalence relations on the manifolds of the domains where we identify two points in a same connected component of a same level set of them. They have been important since the establishment of theory of Morse functions, in the 20th century, They are in certain tame situations $0$- or $1$-dimensional and graphs naturally. Related facts have been shown by Gelbukh and Saeki in the 2020s for certain proper smooth real-valued functions. In non-proper cases, even related explicit theory has been difficult, except some previously given case of the author. Our study is on a new related case.

Given an ideal $\mathcal I$ on $\omega$, a subgroup $H$ of the unit circle $\mathbb T$ is said to be $\mathcal I$-characterized if there exists a sequence of integers $(a_n:n\in\omega)$ such that $$ H= \left\{ x\in\mathbb T: \mathcal I\text{-}\lim_{n\to\infty}a_nx=0 \right\}. $$ We investigate the descriptive complexity and Polishability of these subgroups in terms of the structural and topological properties of the ideal $\mathcal I$. Our main structural result shows that $\mathcal{I}$ is an analytic $P$-ideal if and only if all $\mathcal I$-characterized subgroups are Polishable. In such case, we explicitly describe a compatible finer Polish group topology. Using results on Polishable subgroups, we obtain a trichotomy for their possible Borel complexities. If $\mathcal{I}$ is a generalized density ideal, we show the sharper dichotomy that every proper $\mathcal I$-characterized subgroup is either countable or $F_{\sigma\delta}$-complete. We also prove that this fails for general analytic $P$-ideals by constructing a subgroup characterized by a summable ideal which is neither $F_\sigma$ nor $F_{\sigma\delta}$-complete. Finally, we give explicit descriptions of the subgroups associated with the sequences of powers, the Fibonacci sequence, and the sequence of factorials. We conclude with several open questions.

Let \(\DCPO\) be the category of dcpos and Scott-continuous maps, and let \(\Rtwo\) be the reflective hull of the two-element chain \(2\). We prove that \(\Rtwo\) is the least non-discrete proper reflective full subcategory of \(\DCPO\). It is closed under limits and Skula-closed sub-dcpos and contains every sober dcpo. We show that \(\Rtwo\) is the maximal \(\Gamma\)-faithful full subcategory of \(\DCPO\) that strictly contains weakly dominated dcpos. Finally, we give the concrete construction of \(2\)-reflection internally by a transfinite iteration and a criterion for reflective full subcategories of \(\DCPO\), analogous to the Keimel–Lawson conditions for \(T_0\) topological spaces.