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If $H$ is a Hilbert space, the non-compact Stiefel manifold $St(n,H)$ consists of independent $n$-tuples in $H$. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that $\bigcap_{j \in J} \left( U(j) + St(n,H) \right)$ is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating $J$-family. Then, in the second part, we show that the topological closure of $St(n,H) \cap S$ contains all polynomial paths contained in $S$ and passing through a point in $St(n,H)$. As a consequence, we prove that $St(n,H)$ is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since $St(n,L^2(X,\mu;\mathbb{F}))$ is isometric to $\mathcal{F}_{(X,\Sigma,\mu),n}^\mathbb{F}$, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames., Comment: 25 pages

A $$\sigma $$ σ -ideal $$\mathcal {I}$$ I on a Polish group $$(X,+)$$ ( X , + ) has the Smital Property if for every dense set D and a Borel $$\mathcal {I}$$ I -positive set B the algebraic sum $$D+B$$ D + B is a complement of a set from $$\mathcal {I}$$ I . We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are $$\mathfrak {c}$$ c many maximal invariant $$\sigma $$ σ -ideals with Borel bases on the Cantor space $$2^\omega $$ 2 ω .

In this paper, we deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case -- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic order, and gives a characterisation of the so called weakly cyclically orderable spaces.

Given a continuous and isometric action of a Polish group G on an adequate Polish topometric space $(X,\tau ,\rho )$ and $x \in X$ , we find a necessary and sufficient condition for $\overline {Gx}^{\rho }$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.

Given a completely metrizable space $X$, let $\mathfrak{par}(X)$ denote the smallest possible size of a partition of $X$ into Polish spaces, and $\mathfrak{cov}(X)$ the smallest possible size of a covering of $X$ with Polish spaces. Observe that $\mathfrak{cov}(X) \leq \mathfrak{par}(X)$ for every $X$, because every partition of $X$ is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality $\mathfrak{cov}(X) < \mathfrak{par}(X)$ can hold for some completely metrizable space $X$. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if $\mathfrak{cov}(X) < \mathfrak{par}(X)$ for any completely metrizable $X$, then $0^\dagger$ exists.

Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($\mathcal{I}$), where $\mathcal{I}$ is an ideal on $\omega$, if for each sequence $(x_n)_{n\in\omega}$ in $X$ one can find an $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ converges in $X$. We define an ideal $\mathcal{BI}$ which is critical for FinBW($\mathcal{I}$) in the following sense: Under CH, for every ideal $\mathcal{I}$, $\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Kat\v{e}tov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $\omega_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{\Pi^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($\sigma$-centered) that for $\bf{\Pi^0_4}$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$. Moreover, answering in negative a question of M. Hru\v{s}\'ak and D. Meza-Alc\'antara, we show that the ideal $\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for extendability to a $\bf{\Pi^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.

Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G| the one-complex associated with $G$. The topological space |G| is n-arc connected (n-ac) if every set of no more than n points of |G| are contained in an arc (a homeomorphic copy of the closed unit interval). For any graph G, we show the following are equivalent: (i) |G| in 7-ac, (ii) |G| is n-ac for all n, and (iii) G is a subdivision of one of nine graphs. A graph G has |G| 6-ac if and only if either G is one of the nine 7-ac graphs, or, after suppressing all degree-2-vertices, the graph G is 3-regular, 3-connected, and removing any 6 edges does not disconnect G into 4 or more components. Similar combinatorial characterizations of graphs G such that |G| is n-ac for n=3, 4 and 5 are given. Together these results yield a complete classification of n-ac graphs, for all n.

The lifting problem for continuous bi-equivariant maps and bi-equivariant covering homotopies is considered, which leads to the notion of a bi-equivariant fibration. An intrinsic characteristic of a bi-equivariant Hurewicz fibration is obtained. Theorems concerning a relationship between bi-equivariant fibrations and fibrations generated by them are proved., 8 pages

We prove that a Tychonoff space $X$ is an Ascoli space (resp., a sequentially Ascoli space) if and only if for each Banach space $E$, every $k$-continuous and almost $k$-compact (resp., almost $k$-sequential) map $T$ form $X$ into the Banach dual $E'$ of $E$ is continuous.

The bounded localization $\beta_b$ of a locally convex topology $\beta$ is defined as the finest locally convex topology agreeing with $\beta$ on all bounded sets. We show that the strict topology on the multiplier algebra of a bornological pro-$C^*$-algebras equals its own localization, generalizing the analogous result due to Taylor for multiplier algebras of plain $C^*$-algebras. We also (a) characterize the barreled commutative unital pro-$C^*$-algebras as those of continuous functions on functionally Hausdorff spaces whose relatively pseudocompact subsets are relatively compact, equipped with the topology of uniform convergence on compact subsets, and (b) describe a contravariant equivalence between the category of commutative unital pro-$C^*$-algebras and a category of Tychonoff (rather than functionally Hausdorff) topological spaces., Comment: 22 pages + references

Orbits and bi-invariant subsets of binary $G$-spaces are studied. The problem of the distributivity of a binary action of a group $G$ on a space $X$, which was posed in 2016 by one of the authors, is solved., 11 pages

A topological space is called a submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.

The aim of this paper is to generalize the results on expansive mappings of Yesilkaya and Aydin from \cite{Yesilkaya}. We give some fixed point results for q-expansive mappings in metric spaces and prove some fixed point theorems for this class of mappings. Finally, we present some examples to support the new results.

A compact set has computable type if any homeomorphic copy of the set which is semicomputable is actually computable. Miller proved that finite-dimensional spheres have computable type, Iljazovi\'c and other authors established the property for many other sets, such as manifolds. In this article we propose a theoretical study of the notion of computable type, in order to improve our general understanding of this notion and to provide tools to prove or disprove this property. We first show that the definitions of computable type that were distinguished in the literature, involving metric spaces and Hausdorff spaces respectively, are actually equivalent. We argue that the stronger, relativized version of computable type, is better behaved and prone to topological analysis. We obtain characterizations of strong computable type, related to the descriptive complexity of topological invariants, as well as purely topological criteria. We study two families of topological invariants of low descriptive complexity, expressing the extensibility and the null-homotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones.

For a topological space X, let (RX)s := (RX,Ts) be the cartesian product of |X| copies of the real line R with the topology of the uniform convergence on separable subsets of X. In this article we analyze the subspace C(X) of (RX)s of all real-valued continuous functions on X, denoted by Cs(X). We determine when Cs(X) is dense and when is closed in (RX)s, and we obtain some results about the Baire property in Cs(X). Finally, we determine the cellularity of Cs([0,{\alpha}]) where [0,{\alpha}] is the space of ordinal numbers belonging to {\alpha} + 1 with its usual order topology., Comment: 17 pages

We develop a simple method of constructing topological spaces from countable posets with finite levels, one which applies to all second countable T_1 compacta. This results in a duality amenable to building such spaces from finite building blocks, essentially an abstract analog of classical constructions defining compacta from progressively finer open covers.

We first consider the class K of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for K made up of countable graphs, comparing them with the quasi-order associated with injective continuous homomorphisms. We prove that the size of such a basis is sharp, using odometers. However, using odometers again, we prove that there is no antichain basis in K, and provide infinite descending chains in K. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of the Cantor space is Borel reducible to the equivalence relation associated with our quasi-order. We also prove that there is no antichain basis in the class of graphs on a zero-dimensional Polish space with continuous chromatic number at least three. We study the graphs induced by a continuous function, and show that any basis for the class of graphs induced by a homeomorphism of a zero-dimensional metrizable compact space with continuous chromatic number at least three must have size continuum, using odometers or subshifts.

We study the problem of Nadler and Quinn from 1972, which asks whether, given an arc-like continuum $X$ and a point $x \in X$, there exists an embedding of $X$ in $\mathbb{R}^2$ for which $x$ is an accessible point. We develop the notion of a radial departure of a map $f \colon [-1,1] \to [-1,1]$, and establish a simple criterion in terms of the bonding maps in an inverse system on intervals to show that there is an embedding of the inverse limit for which a given point is accessible. Using this criterion, we give a partial affirmative answer to the problem of Nadler and Quinn, under some technical assumptions on the bonding maps of the inverse system., 23 pages, 8 figures

McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames.

A $graph$ $metric$ on a set $X$ is any function $d: E_d \to\mathbb R_+:=\{x\in\mathbb R:x>0\}$ defined on a connected graph $ E_d \subseteq[X]^2:=\{A\subseteq X:|A|=2\}$ and such that for every $\{x,y\}\in E_d$ we have $d(\{x,y\})\le\hat d(x,y):=\inf\big\{\sum_{i=1}^nd(\{x_{i-1},x_i\}):\{x,y\}=\{x_0,x_n\}\;\wedge\;\{\{x_{i-1},x_i\}:0\check d(x,y:= \sup\{d(\{a,b\})-\hat d(a,u)-\hat d(b,y):\{a,b\}\in E_d \}$ for every $x,y\in X$ with $\{x,y\}\notin E_d $. We prove that for every floppy graph metric $d: E_d \to\mathbb R_+$ on a set $X$, every points $x,y\in X$ with $\{x,y\}\notin E_d $, and every real number $r$ with $\frac 13\check d(x,y)+\frac23\hat d(x,y)\le r, Comment: 18 pages

Universal representation of geometric patterns of disordered matters is investigated with the aid of general topology. By utilizing the result obtained in the previous study (S. Ohmori, et.al., Phys. Scr. 94, 105213 (2019)) that any patterns can be represented by a specific topological space, a construction of topological representation of patterns using Cantor set is shown. The obtained topological representations are then demonstrated by the contractions that characterize the self-similarity of Cantor set. For some practical geometric patterns, e.g., network, dendritic, and clusterized patterns, their topological representations are focused on.

We establish relationships between various topological selection games involving the space of minimal cusco maps into the real line and the underlying domain. These connections occur across different topologies, including the topology of pointwise convergence and the topology of uniform convergence on compacta. Full and limited-information strategies are investigated. The primary games we consider are Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk's closed discrete selection game, and Gruenhage's W-games. We also comment on the difficulty of generalizing the given results to other classes of functions., 23 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2206.13755

Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus., 20 pages

Let $\mathcal{M}(X,\mathcal{A})$ be the ring of all real valued measurable functions defined over the measurable space $(X,\mathcal{A})$. Given an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ and a measure $\mu:\mathcal{A}\to[0,\infty]$, we introduce the $U_\mu^I$-topology and the $m_\mu^I$-topology on $\mathcal{M}(X,\mathcal{A})$ as generalized versions of the topology of uniform convergence or the $U$-topology and the $m$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively. With $I=\mathcal{M}(X,\mathcal{A})$, these two topologies reduce to the $U_\mu$-topology and the $m_\mu$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively, already considered before. If $I$ is a countably generated ideal in $\mathcal{M}(X,\mathcal{A})$, then the $U_\mu^I$-topology and the $m_\mu^I$-topology coincide if and only if $X\setminus \bigcap Z[I]$ is a $\mu$-bounded subset of $X$. The components of $0$ in $\mathcal{M}(X,\mathcal{A})$ in the $U_\mu^I$-topology and the $m_\mu^I$-topology are realized as $I\cap L^\infty(X,\mathcal{A},\mu)$ and $I\cap L_\psi(X,\mathcal{A},\mu)$ respectively. Here $L^\infty(X,\mathcal{A},\mu)$ is the set of all functions in $\mathcal{M}(X,\mathcal{A})$ which are essentially $\mu$-bounded over $X$ and $L_\psi(X,\mathcal{A},\mu)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},\mu)\}$. It is established that an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ is dense in the $U_\mu$-topology if and only if it is dense in the $m_\mu$-topology and this happens when and only when there exists $Z\in Z[I]$ such that $\mu(Z)=0$. Furthermore, it is proved that $I$ is closed in $\mathcal{M}(X,\mathcal{A})$ in the $m_\mu$-topology if and only if it is a $Z_\mu$-ideal in the sense that if $f\equiv g$ almost everywhere on $X$ with $f\in I$ and $g\in\mathcal{M}(X,\mathcal{A})$, then $g\in I$.

The notions of a local $(k_0,k_1)$-isomorphism and a weakly local $(k_0,k_1)$-isomorphism play crucial roles in developing a digital $(k_0,k_1)$-covering space and a pseudo-$(k_0,k_1)$-covering space, respectively. In relation to the study of pseudo-$(k_0,k_1)$-covering spaces, since there are some works to be refined and improved in the literature, the recent paper \cite{H10} improved and corrected some mistakes occurred in the literature. One of the important things is that the notion of a pseudo-$(k_0,k_1)$-covering map in \cite{H6,H9} was revised to be more broadened in \cite{H10}. Thus this new version is proved to be equivalent to a weakly local $(k_0,k_1)$-isomorphic surjection \cite{H10}. The present paper contains some works in \cite{H10} and we only deals with $k$-connected digital images $(X, k)$., Since the paper contains some improvements on covering spaces and pseudocovering spaces, some people can be interesting

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space X=(X,T) is countably compact iff it is countably subcompact relative to T. (iii) For every metric space X=(X,d), X is compact iff it is subcompact relative to T. We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable space is completely metrizable, (b) every countably subcompact metrizable space is subcompact, (c) every complete metrizable space is subcompact is relatively consistent with ZF.

A subset $E$ of a metric space $X$ is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some $n$, sending $E$ to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each $i$ and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space $X$ is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length $N$. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space $X$ shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman-Starbird and Freedman and is applicable to the proof of Freedman's celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological $4$-manifolds, including the $4$-dimensional Poincar\'{e} conjecture., Comment: 11 pages, 2 figures

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016).

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

We investigate a few aspects of the notion of Levi core, introduced by the authors in a previous work: a basic finiteness question, the connections with Kohn's algorithm and with Catlin's property (P).

We define the notion of ideal convergence for sequences $(x_n)$ with values in topological spaces $X$ with respect to a family $\{F_\eta: \eta \in X\}$ of subsets of $X$ with $\eta \in F_\eta$. Each set $F_\eta$ quantifies the degree of accuracy of the convergence toward $\eta$. After proving that this is really a new notion, we provide some properties of the set of limit points and characterize the latter through the ideal cluster points and the ideal core of $(x_n)$.

The theorem we prove is a slight strengthening of some results by Just, Miller, Scheepers and Szeptycki [JMSS]. We use the Michael technique instead of the combinatorial approach in the literature. Comments by the submitter: This short unpublished paper, written in 2002, is a cornerstone in the study of selection principles and classic covering properties. It is the first to solve the Hurewicz Problem, by showing that there is, in ZFC, a real set with Menger's covering property (indeed, in all finite powers) but without Hurewicz's covering property. Unfortunately, web searches begin to miss this paper, and I sought, and received, permission from the authors to store this paper in the ArXiv, providing it a permanent link. The author's decision to not publish this paper, in light of some later stronger results by other authors (including myself), is a true act of generosity. Hopefully, this submission does some justice to their achievement.

The Whitehead Model of free groups can be used to measure the complexity, or degree, of automorphisms of free groups. The bound for the degree of the $f \circ g$ for deg$(f) =$ deg($g) = 0$ had previously been discovered. We extend this result to the case where at least one of our automorphisms has degree 0.

A metric space $(X,d)$ is called a $subline$ if every 3-element subset $T$ of $X$ can be written as $T=\{x,y,z\}$ for some points $x,y,z$ such that $d(x,z)=d(x,y)+d(y,z)$. By a classical result of Menger, every subline of cardinality $\ne 4$ is isometric to a subspace of the real line. A subline $(X,d)$ is called an $n$-$subline$ for a natural number $n$ if for every $c\in X$ and positive real number $r\in d[X^2]$, the sphere $S(c;r):=\{x\in X:d(x,c)=r\}$ contains at least $n$ points. We prove that every $2$-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup $G\subseteq\mathbb R$, a metric space $(X,d)$ is isometric to $G$ if and only if $X$ is a $2$-subline with $d[X^2]=G_+:= G\cap[0,\infty)$. A metric space $(X,d)$ is called a $ray$ if $X$ is a $1$-subline and $X$ contains a point $o\in X$ such that for every $r\in d[X^2]$ the sphere $S(o;r)$ is a singleton. We prove that for a subgroup $G\subseteq\mathbb Q$, a metric space $(X,d)$ is isometric to the ray $G_+$ if and only if $X$ is a ray with $d[X^2]=G_+$. A metric space $X$ is isometric to the ray $\mathbb R_+$ if and only if $X$ is a complete ray such that $\mathbb Q_+\subseteq d[X^2]$. On the other hand, the real line contains a dense ray $X\subseteq\mathbb R$ such that $d[X^2]=\mathbb R_+$., 8 pages

Following Will Brian, we define a metric space $X$ to be $Banakh$ if all nonempty spheres of positive radius $r$ in $X$ have cardinality $2$ and diameter $2r$. Standard examples of Banakh spaces are subgroups of the real line. In this paper we study the geometry of Banakh spaces, characterize Banakh spaces which are isometric to subgroups of the real line, and also construct Banakh spaces $(X,d)$ which do not embed into the real line and have a prescribed distance set $d[X^2]$., 40 pages

In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. The main result of this paper is that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism group of the free product of four finite groups or the free product of a single finite group with a free group of rank two is a virtual duality group of dimension two, in contrast with the above example. We also prove that groups in this family are semistable at infinity (or at each end). Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krsti\'c and Vogtmann., Comment: 27 pages, 2 figures

Answering a question of Garbuli\'nska-W\c{e}grzyn and Kubi\'s, we prove that Gurarii operators form a dense $G_\delta$-set in the space $\mathcal B(\mathbb G)$ of all nonexpansive operators on the Gurarii space $\mathbb G$, endowed with the strong operator topology. This implies that the set of universal operators on $\mathbb G$ form a residual set in $\mathcal B(\mathbb G)$., Comment: 12 pages

We discuss topological rigidity of vector bundles with asymptotically conical (AC) total spaces of rank greater than 1 with a sufficiently connected link; our focus will mainly be on ALE (asymptotically locally Euclidean) bundles. Within the smooth category, we topologically classify all ALE tangent bundles by showing only 2-sphere, projective plane and open contractible manifolds admit ALE tangent bundles. We also discuss other interesting topological and geometric rigidities of ALE vector bundles., 14 pages, no figures

We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called balancing condition if and only if it is an indexed branched cover, i.e.~locally over connected sets the count with multiplicity of points in every fibre is a constant, which in particular gives a well-defined global degree when the target is connected. Given a balanced map $(\Sigma, m_\Sigma) \to \Delta$, we lift several connectivity properties of $\Delta$ to~$\Sigma$. Using these lifting results we determine whether a multiplicity $m_{\mathcal U}$ that is defined only on the interiors of maximal cells of $\Sigma$ can be extended to all $\Sigma$ in a balanced manner. This relies on a strong connectivity assumption; we give a counterexample when this is missing., Comment: 34 pages, 9 figures, comments are welcome!

Suppose that $S_1$ and $S_2$ are nonempty subsets of a complete metric space $(\mathcal{M},d)$ and $\phi,\psi:S_1\to S_2$ are mappings. The aim of this work is to investigate some conditions on $\phi$ and $\psi$ such that the two functions, one that assigns to each $x\in S_1$ exactly $d(x,\phi x)$ and the other that assigns to each $x\in S_1$ exactly $d(x,\psi x)$, attain the global minimum value at the same point in $S_1$. We have introduced the notion of proximally $F$-weakly dominated pair of mappings and proved two theorems that guarantee the existence of such a point. Our work is an improvement of earlier work in this direction. We have also provided examples in which our results are applicable, but the earlier results are not applicable.

M. Escard\'o et al. asked whether the core compactly generated topology of a sober space is again sober and the sobrification of a core compactly generated space again core compactly generated. In this note, we answer the problem by displaying a counterexample, which reveals that the core compactly generated spaces are not closed under sobrifications. Meantime, we obtain that the core compactly generated spaces are closed under {\omega}-well-filterifications and D-completions. Furthermore, we find that the core compactly generated topology of the Smyth power space of a well-filtered space coincides with the Scott topology. Finally, we provide a characterization for the core-compactness of core compactly generated spaces., Comment: 11 pages, 1 figure

Stone space partitions $\{X_{p}\mid p\in P\}$ satisfying conditions like $\bar{X_{p}}=\bigcup_{q\leqslant p}X_{q}$ for all $p\in P$, where $P$ is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of $\omega$-categorical structures. A key concept for studying such partitions is that of a $p$-trim open set which meets precisely those $X_{q}$ for which $q\geqslant p$; for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition $\{X_{r}\mid r\in[0,1]\}$ of the Cantor set such that $\bar{X_{r}}=\bigcup_{s\leqslant r}X_{s}\text{ for all }r\in[0,1]$., Comment: 41 pages

This paper presents a framework for semantic segmentation on sparse sequential point clouds of millimeter-wave radar. Compared with cameras and lidars, millimeter-wave radars have the advantage of not revealing privacy, having a strong anti-interference ability, and having long detection distance. The sparsity and capturing temporal-topological features of mmWave data is still a problem. However, the issue of capturing the temporal-topological coupling features under the human semantic segmentation task prevents previous advanced segmentation methods (e.g PointNet, PointCNN, Point Transformer) from being well utilized in practical scenarios. To address the challenge caused by the sparsity and temporal-topological feature of the data, we (i) introduce graph structure and topological features to the point cloud, (ii) propose a semantic segmentation framework including a global feature-extracting module and a sequential feature-extracting module. In addition, we design an efficient and more fitting loss function for a better training process and segmentation results based on graph clustering. Experimentally, we deploy representative semantic segmentation algorithms (Transformer, GCNN, etc.) on a custom dataset. Experimental results indicate that our model achieves mean accuracy on the custom dataset by $\mathbf{82.31}\%$ and outperforms the state-of-the-art algorithms. Moreover, to validate the model's robustness, we deploy our model on the well-known S3DIS dataset. On the S3DIS dataset, our model achieves mean accuracy by $\mathbf{92.6}\%$, outperforming baseline algorithms.

The celebrated Assouad embedding theorem has been known for over 40 years. It states that for any doubling metric space (with doubling constant $C_0$) there exists an integer $N$, such that for any $\alpha\in (0.5,1)$ there exists a positive constant $C(\alpha,C_0)$ and an injective function $F:X\to \mathbb{R}^N$ such that \[ \forall x,y\in X \quad C^{-1} d(x,y)^{\alpha} \leq \|F(x)-F(y) \| \leq Cd(x,y)^{\alpha} \] In the paper we use the remetrization techniques to extend the said theorem to a broad subclass of semimetric spaces. We also present the limitations of this extension -- in particular, we prove that in any semimetric space which satisfies the claim of the Assouad theorem, the relaxed triangle condition holds as well, which means that there exists $K\geq 1$ such that \[ \forall x,y,z\in X \quad d(x,z)\leq K\left( d(x,y)+d(y,z)\right). \], Comment: 9 pages

We prove that the classes of weakly $1$-dimensional and almost $0$-dimensional spaces are disjoint. The result has applications to hereditarily locally connected spaces, $\mathbb R$-trees, and endpoints of smooth fans.

We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting behaviour of slender groups. Secondly, we remove the condition of being abelian and consider nilpotent periodic locally compact $p$-groups ($p$ prime), reducing the computations to the case of Sylow $p$-subgroups. Finally, we investigate locally compact Heisenberg $p$-groups $\mathbb{H}_{n}(\mathbb{Q}_{p})$ on the field $\mathbb{Q}_{p}$ of the $p$-adic rationals with $n$ arbitrary positive integer., 11 pages ; 1 figure

Let $\mathcal{P}$ be an ideal of closed subsets of a topological space $X$. Consider the ring, $C(X)_\mathcal{P}$ of real valued functions on $X$ whose closure of discontinuity set is a member of $\mathcal{P}$. We investigate the ring properties of $C(X)_\mathcal{P}$ for different choices of $\mathcal{P}$, such as the $\aleph_0$-self injectivity and regularity of the ring, if and when the ring is Artinian and/or Noetherian. The concept of $\mathcal{F}P$-space was introduced by Z. Gharabaghi, M. Ghirati and A. Taherifar in 2018 in a paper published in Houston Journal of Mathematics. In this paper, they established a result stating that every $P$-space is a $\mathcal{F}P$-space. We furnish that this theorem might fail if $X$ is not Tychonoff and we provide a suitable counter example to prove our assertion.

3-D shape is important to chemistry, but how important? Machine learning works best when the inputs are simple and match the problem well. Chemistry datasets tend to be very small compared to those generally used in machine learning so we need to get the most from each datapoint. Persistent homology measures the topological shape properties of point clouds at different scales and is used in topological data analysis. Here we investigate what persistent homology captures about molecular structure and create persistent homology features (PHFs) that encode a molecule's shape whilst losing most of the symbolic detail like atom labels, valence, charge, bonds etc. We demonstrate the usefulness of PHFs on a series of chemical datasets: QM7, lipophilicity, Delaney and Tox21. PHFs work as well as the best benchmarks. PHFs are very information dense and much smaller than other encoding methods yet found, meaning ML algorithms are much more energy efficient. PHFs success despite losing a large amount of chemical detail highlights how much of chemistry can be simplified to topological shape., 18 pages, 15 figures

This book provides an introduction to the theory of digital (molecular) spaces (TDS). Digital spaces are combinatorial models of continuous spaces. TDS is one of alternative branches of digital topology that studies constructing and modifying 2, 3 and n-dimensional digital image arrays in a computer and its memory. Demands for mathematical theory of multidimensional spaces built of finite number of points appeared in connection with the use of many dimensional images in computer programs. This work presents, as examples, digital models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus, a projective plane, etc. Methods of TMS can work successfully in such areas as biology, chemistry, industry, medicine, etc. The book is organized as follows. Digital models of continuous spaces are the intersection graphs of special locally centered lamp covers of continuous spaces. Digital models are represented by graphs, algebraic matrices and a set of unit cubes in n-dimensional Euclidean space. The main mathematical characteristics of digital models of continuous spaces, such as the dimension, the Euler characteristic, the homology groups and others, are the same as those of their continuous counterparts. Contractible transformations of digital spaces, that change the number of elements, do not change the mathematical characteristics and properties of digital spaces. The main goal is to construct digital models of continuous spaces and to explain how to use the new methods to work with mathematical objects. The emphasis is on introducing the readers to the basics and conceptual understanding of what TDS is., 345 pages, in Russian

Previously, we have systematically constructed explicit real algebraic functions which are represented as the compositions of smooth real algebraic maps whose images are domains surrounded by hypersurfaces of degree 1 or 2 with canonical projections. Here we give new examples with the hypersurfaces each of which is the product of a connected component of a hyperbola and a copy of the $1$-dimensional affine space explicitly. As a related future work we also discuss problems to obtain the zero sets of some real polynomials explicitly from increasing sequences of real numbers. This is motivated by a problem in theory of smooth functions proposed first by Sharko: can we construct nice smooth functions whose Reeb graphs are as prescribed? The Reeb space of a smooth function is the naturally obtained graph whose underlying space is the quotient space of the manifold consisting of connected components of preimages. The author first considered variants respecting the topologies of the preimages and obtained several results before. Our work is also motivated by real algebraic geometry, pioneered by Nash. We can know existence of real algebraic structures of smooth manifolds and some general sets and we already know several approximations of smooth maps by real algebraic maps. Our interest lies in explicit construction, which is difficult., 11 pages, this overlaps arXiv:2303.14988, which has been withdrawn, and arXiv:2304.02372, giving another explicit answer on construction of explicit real algebraic maps