One-point Compactification Research Articles

One-point connectifications of regular spaces

It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the Alexandroff compactification) if and only if it is non-compact. There is also, an old question of Alexandroff of characterizing spaces which have a one-point connectification. Here, we study one-point connectifications in the realm of regular spaces and prove that a locally connected space has a regular one-point connectification if and only if the space has no regular-closed component. This, also gives an answer to the conjecture raised by M. R. Koushesh. Then, we consider the set of all one-point connectifications of a locally connected regular space and show that, this set (naturally partially ordered) is a compact conditionally complete lattice. Further, we extend our theorem for locally connected regular spaces with a topological property P and give conditions on P which guarantee the space to have a regular one-point connectification with P.

Hereditarily projectable Archimedean lattice-ordered groups with unit

Let W be the category of Archimedean lattice-ordered groups with distinguished weak unit and unit-preserving ℓ-group homomorphisms. We denote by P, wP, Loc, HA, respectively, the collections of projectable, weakly projectable, local, and hyperarchimedean W-objects. For any C ⊆ W, we say that a W-object H is hereditarily C, denoted H ∈ hC, if G ∈ C whenever G is a W-subobject of H. Our focus is hP. Taking off from the known items P = wP ∩ Loc, h HA = HA ⊆ P, and C (X) ∈ P iff X is basically disconnected (BD), we show For C (X), equivalent are: hP; hLoc and X BD; X finite. If the Yosida space Y G is BD, then (1) If G is hLoc, then G is bounded; (2) If G is hP, then G is HA. hP = hwP, and every G ∈ hP with G ∉ HA has Yosida space Y G = αN, the one-point compactification of N. There are various G ∈ hP with Y G = αN, including: G bounded not HA; G not bounded with bounded part HA; G not bounded with bounded part not HA.

Soft nodec spaces

<abstract><p>Following van Douwen, we call a soft topological space soft nodec if every soft nowhere dense subset of it is soft closed. This paper considers soft nodec spaces, which contain soft submaximal and soft door spaces. We investigate the basic properties and characterizations of soft nodec spaces. More precisely, we show that a soft nodec space can be written as a union of two disjoint soft closed soft dense (or soft open) soft nodec subspaces. Then, we study the behavior of soft nodec spaces under various operations, including the following: taking soft subspaces, soft products, soft topological sums, and images under specific soft functions with the support of appropriate counterexamples. Additionally, we show that the Krull dimension of a soft nodec soft $ T_{0} $-space is less than or equal to one. After that, we present some connections among soft nodec, soft strong nodec, and soft compact spaces. Finally, we successfully determine a condition under which the soft one-point compactification of a soft space is soft nodec if and only if the soft space is soft strong nodec.</p></abstract>

A geometric approach to equivariant factorization homology and nonabelian Poincaré duality

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an $$\textrm{E}_V$$ -algebra using a two-sided bar construction, generalizing (Andrade, From manifolds to invariants of $$E_n$$ -algebras. PhD thesis, Massachusetts Institute of Technology, 2010; Kupers and Miller, Math Ann 370(1–2):209–269, 2018). This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincaré duality theorem due to several authors. The eNPD states that the scanning map gives a G-equivalence from the equivariant factorization homology to mapping spaces out the one-point compactification of the G-manifolds, when the coefficients are G-connected. For non-G-connected coefficients, when the G-manifolds have suitable copies of $$\mathbb {R}$$ in them, the scanning map gives group completions. This generalizes the recognition principle for V-fold loop spaces in Guillou and May (Algebr Geom Topol 17(6):3259–3339, 2017).

On c-sober spaces and ω*-well-filtered spaces

Based on countably irreducible version of Topological Rudin?s Lemma, we give some characterizations of c-sober spaces and ?*-well-filtered spaces. In particular, we prove that a topological space is c-sober iff its Smyth power space is c-sober and a c-sober space is an ?*-well-filtered space. We also show that a locally compact ?+-well-filtered P-space is c-sober and a T0 space X is c-sober iff the one-point compactification of X is c-sober.

Conformally invariant complete metrics

AbstractFor a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [35]. Next, we prove that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.

Koszul duality for topological E n ‐operads

We show that the Koszul dual of an E n $E_n$ -operad in spectra is O ( n ) $O(n)$ -equivariantly equivalent to its n $n$ -fold desuspension. To this purpose we introduce a new O ( n ) $O(n)$ -operad of Euclidean spaces R n $R_n$ , the barycentric operad, that is fibred over simplexes and has homeomorphisms as structure maps; we also introduce its suboperad of restricted little n $n$ -discs D n $D_n$ , that is an E n $E_n$ -operad. The duality is realised by an unstable explicit S-duality pairing ( F n ) + ∧ B D n → S ¯ n $(F_n)_+ \wedge BD_n \rightarrow \bar{S}_n$ , where B $B$ is the bar-cooperad construction, F n $F_n$ is the Fulton–MacPherson E n $E_n$ -operad, and the dualising object S ¯ n $\bar{S}_n$ is an operad of spheres that are one-point compactifications of star-shaped neighbourhoods in R n $R_n$ . We also identify the Koszul dual of the operad inclusion map E n → E n + m $E_n \rightarrow E_{n+m}$ as the ( n + m ) $(n+m)$ -fold desuspension of an unstable operad map E n + m → Σ m E n $E_{n+m} \rightarrow \Sigma ^m E_n$ defined by May.

Topological Entropy of Free Semigroup Actions Generated by Proper Maps for Noncompact Subsets

In this paper, we introduce three notions of topological entropy of a free semigroup action generated by proper maps for noncompact subsets, which extends the notions defined by Ju et al. [13] and Ma et al. [17]. By using the one-point compactification as a bridge, we study the relations of the entropies between two dynamical systems. We then introduce three skew-product transformations, and for a particular subset, the relationship between the upper capacity topological entropy of a free semigroup action generated by proper maps, and the upper capacity topological entropy of a skew-product transformation is given. As applications, we examine the multifractal spectrum of a locally compact separable metric space, and it is shown that the irregular set has full upper capacity topological entropy of a free semigroup action generated by proper maps.

Semisolid sets and topological measures

This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is defined on open and closed sets; it is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. We examine semisolid sets and give a way of constructing topological measures from solid-set functions on locally compact, Hausdorff, connected, locally connected spaces. For compact spaces we present two simpler methods than the currently available one. We give examples of finite and infinite topological measures on locally compact spaces and present an easy way to generate topological measures on spaces whose one-point compactification has genus 0. Our results are necessary for various methods for constructing topological measures, give additional properties of topological measures, and provide a tool for determining whether two topological measures or quasi-linear functionals are the same.

Derivations on the algebras of the one-point compactification of affine semigroups

For any affine semigroup S, the set has a natural semigroup structure; additionally, if S is endowed with the discrete topology, then the semigroup can be studied as the one-point compactification of S. In this article, we study the derivations on semigroup algebra in relation to the derivations on semigroup algebra considering the metrizable topology on induced by the one-point compactification topology of

Existence, uniqueness and ergodic properties for time-homogeneous Itô-SDEs with locally integrable drifts and Sobolev diffusion coefficients

Using elliptic and parabolic regularity results in $L^p$-spaces and generalized Dirichlet form theory, we construct for every starting point weak solutions to SDEs in $\mathbb{R}^d$ up to their explosion times including the following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient $A=(a_{ij})_{1\le i,j\le d}$ is locally uniformly strictly elliptic with functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and the drift coefficient $\mathbf{G}=(g_1,\dots, g_d)$ consists of functions $g_i\in L^p_{loc}(\mathbb{R}^d)$. The solution originates by construction from a Hunt process with continuous sample paths on the one-point compactification of $\mathbb{R}^d$ and the corresponding SDE is by a known local well-posedness result pathwise unique up to an explosion time. Just under the given assumptions we show irreducibility and the strong Feller property on $L^{1}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$ of its transition function, and the strong Feller property on $L^{q}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$, $q=\frac{dp}{d+p}\in (d/2,p/2)$, of its resolvent, which both include the classical strong Feller property. We present moment inequalities and classical-like non-explosion criteria for the solution which lead to pathwise uniqueness results up to infinity under presumably optimal general non-explosion conditions. We further present explicit conditions for recurrence and ergodicity, including existence as well as uniqueness of invariant probability measures.

Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

AbstractWe study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\], endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\]) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\]. As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\].We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\], we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

On feebly compact topologies on the semigroup BF1ω

We study the Gutik-Mykhalenych semigroup B ω F 1 in the case when the family F 1 consists of the empty set and all singleton in ω . We show that B ω F 1 is isomorphic to subsemigroup B ω ↗ ω m i n of the Brandt ω -extension of the semilattice ω , m i n and describe all shift-continuous feebly compact T 1 -topologies on the semigroup B ω ↗ ω m i n . In particular, we prove that every shift-continuous feebly compact T 1 -topology τ on B ω F 1 is compact and moreover in this case the space ( B ω F 1 , τ ) is homeomorphic to the one-point Alexandroff compactification of the discrete countable space D ( ω ) .

The Baire property on the hyperspace of nontrivial convergent sequences

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences Sc(X) of a Fréchet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which Sc(X) is Baire: this kind of spaces have a dense subset of isolated points (see [12]). We characterize the spaces X for which Sc(X) is Baire. This characterization uses a topological game inspired by the Banach-Mazur game. As a consequence, we obtain that if X is completely metrizable and has a dense subset of isolated points, then Sc(X) is Baire. Our last main result shows that Sc(X) is pseudocompact iff X is homeomorphic to the one-point compactification of a discrete space of uncountable size. This last assertion provides a characterization of the one-point compactification of a discrete space of uncountable size.

Function space topologies between the uniform topology and the Whitney topology

This paper, dedicated to new function space topologies between the uniform topology and the Whitney topology also in the setting of the ωμ-metric spaces, splits in two parts. In the former, where X is a Tychonoff space and (Y,d) is a non-discrete metric space, we explore suggestive uniformizable function space topologies on C(X,Y), the set of all continuous functions from X to Y, located between the uniform topology and the Whitney topology. In the Whitney uniformity, whose natural associated topology is the Whitney topology, any continuous function from X to the positive reals gives a measure of closeness between functions in C(X,Y). But, a less stringent and, by the way, efficient uniform control can be performed equally well by limiting, as for example at a first glance, to the measures deriving from all continuous positive functions continuously extendable to a T2-compactification of X. And next, when X is a local proximity space, i.e. densely embedded in a natural T2 local compactification l(X), by limiting to the positive ones in C(l(X),R). We investigate two classes of Tychonoff spaces. That of locally compact ones splittable in two essentially different cases: X hemicompact or not. And, that of spaces densely embedded in a locally compact one. We prove that, whenever X is hemicompact, then any weak Whitney topology relative to a T2-compactification of X agrees with the classical one. Whenever X is locally compact but not hemicompact, then the weak Whitney topology associated with its one-point compactification reduces just to the uniform topology. In the case X is locally compact, paracompact but not hemicompact, thus the free union of an uncountable family of open σ-compact subsets, then, between the uniform topology and the Whitney topology there is a great variety of weak Whitney topologies relative to T2-compactifications of X. Also, whenever X is not locally compact, weak Whitney topologies associated with different T2 local compactifications of X are generally different as is the case if X is the rational Euclidean line. So, weakening the Whitney topology but without renouncing to the uniform convergence, we produce different uniformizable topologies on C(X,Y) related to various significant structures on X.In the latter, since ωμ-metric spaces, where ωμ is an ordinal number, fill a large and attractive class of peculiar uniform spaces containing the usual metric ones, we focus our attention on the ωμ-metric framework. Indeed, we extend the Whitney topology to C(X,Y), where X is again a Tychonoff space but Y is replaced with an ωμ-metric space. Precisely, the range space Y carries a distance ρ:Y×Y→G, sharing the usual formal properties with real metrics but valued in an ordered Abelian additive group G, which admits a strictly decreasing ωμ-sequence converging to zero in the order topology. By a proof strategy essentially based on zero-dimensionality of any ωμ-metric space with μ>0, we achieve, among others, the following result: Whenever X is an ωμ-additive and paracompact space and (Y,ρ,G) is an ωμ-metric space, then the Whitney topology on C(X,Y) is independent of the ωμ-metric ρ. More precisely, the Whitney topology is a topological character as in the classical metric case, μ=0, and X paracompact.

Ultrapowers of topological spaces

We use ultrapowers to build p-compactifications (for a free ultrafilter p on ω) of topological spaces. In particular we show that the universal p-compactification derived from the Stone-Čech compactification, the one-point compactification of a locally compact space and every p-compactification of a discrete space are special cases of this construction. We will also study when the ultrapower of a topological group is a topological group.

Signed Topological Measures on Locally Compact Spaces

In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is τ-smooth on open sets and τ-smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed topological measures, which in turn is properly contained in the family of signed deficient topological measures. Extending known results for compact spaces, we prove that a signed topological measure is the difference of its positive and negative variations if at least one variation is finite; we also show that the total variation is the sum of the positive and negative variations. If the space is locally compact, connected, locally connected, and has the Alexandrov one-point compactification of genus 0, a signed topological measure of finite norm can be represented as a difference of two topological measures.

A generalized Kasner transition for bouncing Bianchi I models in modified gravity theories

We derive transition rules for Kasner exponents in bouncing Bianchi I models with generic perfect fluid matter fields for a broad class of modified gravity theories where cosmological singularities are resolved and replaced by a non-singular bounce. This is a generalization of results obtained previously in limiting curvature mimetic gravity and loop quantum cosmology. A geometric interpretation is provided for the transition rule as a linear map in the Kasner plane. We show that the general evolution of anisotropies in a Bianchi I universe—including during the bounce phase—is equivalent to the motion of a point particle on a sphere, where the sphere is the one-point compactification of the Kasner plane. In addition, we study the evolution of anisotropies in a large family of bouncing Bianchi I space-times. We also present a novel explicit solution to the Einstein equations for a Bianchi I universe with ekpyrotic matter with a constant equation of state ω=3.

The structure of <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets of asymptotically non-autonomous discrete dynamical systems

We consider a discrete non-autonomous semi-dynamical system generated by a family of continuous maps defined on a locally compact metric space. It is assumed that this family of maps uniformly converges to a continuous map. Such a non-autonomous system is called an asymptotically autonomous system. We extend the dynamical system to the metric one-point compactification of the phase space. This is done via the construction of an associated skew-product dynamical system. We prove, among other things, that the omega limit sets are invariant and invariantly connected. We apply our results to two populations models, the Ricker model with no Allee effect and Elaydi-Sacker model with the Allee effect, where it is assumed that the reproduction rate changes with time due to habitat fluctuation.

Spectral primal spaces

Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results.