Non-discrete Space Research Articles

Continuous Selections and Extremally Disconnected Spaces

This paper deals with extremally disconnected spaces and extremally disconnected P-spaces. A space X is said to be extremally disconnected if, for every open subset V of X, the closure of V in X is also an open set. P-spaces are spaces in which the intersection of countably many open sets is an open set. The authors present a new characterization of extremally disconnected spaces, and the extremally disconnected P-spaces, by means of selection theory. If X is either an extremally disconnected space or an extremally disconnected P-space, then the usual theorems of extension of real-valued continuous functions for a dense subset S of X can be deduced from our results. A corollary of our outcomes is that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X.

Non-normality points and nice spaces

J. Terasawa in "$\beta X-\{p\}$ are non-normal for non-discrete spaces $X$" (2007) and the author in "On non-normality points and metrizable crowded spaces" (2007), independently showed for any metrizable crowded space $X$ that each point $p$ of its Čech--Stone remainder $X^{*}$ is a non-normality point of $\beta X$. We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

Geometric property (T) for non-discrete spaces

Geometric property (T) was defined by Willett and Yu, first for sequences of graphs and later for more general discrete spaces. Increasing sequences of graphs with geometric property (T) are expanders, and they are examples of coarse spaces for which the maximal coarse Baum-Connes assembly map fails to be surjective. Here, we give a broader definition of bounded geometry for coarse spaces, which includes non-discrete spaces. We define a generalisation of geometric property (T) for this class of spaces and show that it is a coarse invariant. Additionally, we characterise it in terms of spectral properties of Laplacians. We investigate geometric property (T) for manifolds and warped systems.

Effective velocity-correction projection methods for unsteady incompressible natural convection equations

We present and analyze two categories of efficient decoupled methods for the unsteady incompressible natural convection equations in this article. The proposed methods include first order, second order standard form and rotational form, and are extended to any order schemes. Some consistency items are added to help complete the stability analysis, and they disappear in non-discrete space. Finally, the correctness of the theoretical analysis and the effectiveness of the given methods are verified by several numerical experiments.

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.

A non-discrete space X with Cp(X) Menger at infinity

<p>In a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of C<sub>p</sub>(X) in some compactification is Menger but not σ-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.</p>

A note on discrete C-embedded subspaces

Abstract It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of υ X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of υ X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact.

Biodiversity and climate change adaptation through non-discrete architectural spaces and architectures

The research claims that traditions are not static. They develop and adapt based on the present situation. Due to the recent climate extremes coming to formally mild climate locations, their architectures can learn from traditional ones from more climate extreme locations. The present systemic design study on semi-interior, ‘non-discrete spaces’ (Hensel, 2013; Hensel & Turko, 2015), of Norwegian traditional architectures, so called ‘svalgangs’ and ‘skuts’ examine its reuse for today climate change adaptation and support of biodiversity that is currently decreasing. Our agricultural land become so toxic, that its species are recently moving and adapting for life in the cities. The discussed traditional spaces offer various boundary penetration of its surrounding environment while providing mediation of its biotic and abiotic agency. These do not cover only anthropocentric benefits for its users such as light and climate comfort but also offer opportunities of communication with other species or their sheltering. 
 This practitioners’ historical research survey motivated by design co-developes its own systemic process based methodology Systemic Approach to Architectural Performance that originates from ‘Systems Oriented Design’ (Sevaldson, 2013b) and ‘Time Based Design’ (Sevaldson, 2004). Where, this ‘non-anthropocentric architecture’ (Hensel, 2012) is in over-evolving co-design with ambient environment’s abiotic and biotic agents, including humans.

The existence of continuous weak selections and orderability-type properties in products and filter spaces

Orderability, weak orderability and the existence of continuous weak selections on spaces with a single non-isolated point and their products are discussed. We prove that a closed continuous image X of a suborderable space must be hereditarily paracompact provided that its product X×Y with some non-discrete space Y has a separately continuous weak selection.

Some topologies on the spaces of USCO maps and densely continuous forms

We study the completeness of three (metrizable) uniformities on the sets D(X, Y) and U(X, Y) of densely continuous forms and USCO maps from X to Y: the uniformity of uniform convergence on bounded sets, the Hausdorff metric uniformity and the uniformity U B . We also prove that if X is a nondiscrete space, then the Hausdorff metric on real-valued densely continuous forms D(X, ℝ) (identified with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y) equipped with the Hausdorff metric is dense equicontinuity introduced by Hammer and McCoy in [7].

Two preservation results for countable products of sequential spaces

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Let $\mathcal{A}$ be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of $\mathcal{A}$ . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of $\mathcal{A}$ containing a non-discrete space is generated by a class of prime spaces and if $\mathcal{A}$ is a quotient-reflective subcategory of Top, then the assignment $\mathcal{B}\mapsto \mathcal{B}\cap \mathcal{A}$ gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of $\mathcal{A}$ that contain $\mathbf{FG}\cap \mathcal{A}$ . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.

Two discrete–non-discrete results

The aim of the paper is to prove two DND theorems, that is properties which make sense on both discrete and non-discrete space. The first one regards the image, on a compact metric space, of a surjective semicontraction, that is a Lipschitz function with a constant less than or equal to 1. The second one involves a generalized notion of convexity, and points out that there exist discrete metric spaces, none of whose intervals is trivial.

Network Reliability and Borel's Paradox

Borel's paradox is a paradox in probability theory with implications in modeling and inference. The paradox arises when we induce probabilities from a high-dimensional, nondiscrete space to its lower dimensional subspace using conditioning arguments that are ill defined. The purpose of this article is to draw attention to the fact that when assessing the reliability of a network there are circumstances under which the paradox comes into play, and to show the consequences of the paradox on the calculated reliability. In particular, we are able to establish the counter-intuitive result that the reliability of a series system of n components, whose life-lengths are independent, converges to a half (and not to zero), as n increases.

Pontryagin Duality for Spaces of Continuous Functions

A topological abelian group G is P-reflexive if the natural homomorphism of G to its Pontryagin bidual group is a topological isomorphism. Let Cp(X) be the space of continuous functions with the topology of pointwise convergence. We investigate for what spaces X the group Cp(X) is P-reflexive. We show that: (1) if Cp(X) is P-reflexive, then X is a P-space; (2) there exists a non-discrete space X such that Cp(X) is P-reflexive; (3) there exists a P-space X such that Cp(X) is not P-reflexive; (4) there exists a simple space X for which the question of whether Cp(X) is P-reflexive is undecidable in ZFC.

On field theory quantization around instantons

Various aspects of the so-called topological embedding, a procedure recently proposed for quantizing a field theory around a non-discrete space of classical minima, are discussed and collected in a simple logical scheme. The possible physical implications are pointed out. The compatibility of the procedure with renormalization is illustrated in the case of the Yang - Mills theory expanded around instantons. The quantum topological properties of Yang - Mills instantons are re-derived in a simpler and illustrative way. Moreover, the general approach is applied to the free energy of the Ginzburg - Landau theory of superconductivity in the intermediate situation between type I and type II superconductors. The topological version of the theory is solved and the quantum topological sectors of the static vortices are classified.

Function spaces of low Borel complexity

In this paper we investigate situations in which the space C π ( X ) {C_\pi }(X) of continuous, real-valued functions on X X is a Borel subset of the product space R X {{\mathbf {R}}^X} . We show that for completely regular, nondiscrete spaces, C π ( X ) {C_\pi }(X) cannot be a G δ {G_\delta } , an F σ {F_\sigma } , or a G δ σ {G_{\delta \sigma }} subset of R X {{\mathbf {R}}^X} , but it can be an F σ δ {F_{\sigma \delta }} or G δ σ δ {G_{\delta \sigma \delta }} subset.

On large cohomological dimension and tautness

AbstractWe prove that for a non-discrete spaceX, the inequality DimL(X) ≥ dimL(X) + 1 always holds if (i)Xis paracompact and each point isGδ, or (ii)Xis a completely paracompact Moritak-space. Consequently, ifXis a non-discrete completely paracompact space in which each point is aGδ-set or it is also a Moritak-space then, the equality DimL(X) = dimL(X) + 1 always holds. We apply this equality to show that for such a spaceXthere exists a pointx ∈ Xand a family ϕ of supports onXsuch that {x} is not ϕ-taut with respect to sheaf cohomology. This generalizes a corresponding known result forRn. We also discuss the usual sum theorems for this large cohomological dimension; the finite sum theorem for closed sets is proved, and for all others, counter examples are given. Subject to a small modification, however, all of the sum theorems hold for a large class of spaces.

Category in function spaces. I

In this paper we study Baire category in spaces of continuous, real-valued functions equipped with the topology of pointwise convergence. We show that, for normal spaces, the Baire category of Cπ(X) is determined by the Baire category of Cπ(Y) for certain small subspaces Y of X and that the category of Cπ(X) is intimately related to the existence of winning strategies in a certain topological game Γ(X) played in the space X. We give examples of certain countable regular spaces for which Cπ(X) is a Baire space and we characterize those spaces X for which Cπ(X) has one of the stronger completeness properties, such as pseudocomplet eness or Cech-completeness. l Introduction* It is well-known that the set of all continuous real-valued functions on a space X is a completely metrizable space when equipped with the topology of uniform convergence and therefore that the Baire Category Theorem is valid in this space. However, when function spaces carry other topologies, the status and role of the Baire category property is more mysterious, and in this paper we consider Baire category in Cπ{X), the set of all continuous real-valued functions on a completely Hausdorff space X, equipped with the topology of pointwise convergence. (See § 2 for precise definitions.) We can preview some of our less technical results as follows. Experience shows that the Baire category of Cπ(X) is determined by the kinds of limit points which countable subsets of X can have. For example, if Cπ(X) is a Baire space then X has no nontrivial convergent sequences (Corollary 3.3) and the same argument shows that the set of bounded members of Cπ(X) is never a Baire space. Furthermore, it is sometimes possible to study the Baire category of Cπ(X) by examining the continuously extendable functions defined on countable subspaces of X (3.7). Examples show that there are infinite nondiscrete spaces X for which Cπ(X) is a Baire space, e.g., any normal space in which each countable subset is closed (see Theorem 8.4). A more interesting fact is that there are even countable nondiscrete spaces for which Cπ(X) is a Baire space, e.g., any space X = ω U {p} where p e βω — ω, topologized as a subspace of βω (see Example 7.1). It is possible for Cπ(X) to be a Baire space, where X is countable and regular, without X being embeddable in βω (Example 7.2) and we can characterize those filters ^V on ω such <that Gπ{X) is a Baire space when X is obtained by adjoining a single point to ω and using Λr as the neighborhood filter of the ideal point. More precisely, if X has a unique nonisolated

Semigroups of continuous selfmaps for which Green's and ℐ relations coincide

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following