scholarly journals On spaces whose nowhere dense subsets are scattered

1998 ◽  
Vol 21 (4) ◽  
pp. 735-740
Author(s):  
Julian Dontchev ◽  
David Rose
Keyword(s):  
Topological Space ◽  
Scattered Spaces

The aim of this paper is to study the class ofN-scattered spaces, i.e. the spaces whose nowhere dense subsets are scattered. The concept was recently used in a decomposition of scatteredness — a topological space(X,τ)is scattered if and only ifXisα-scattered (= itsα-topology is scattered) andN-scattered.

scholarly journals Fan-Gottesman Compactification and Scattered Spaces

2020 ◽  
Vol 5 (1) ◽  
pp. 475-478
Author(s):  
Ceren Sultan Elmali ◽  
Tamer Ugŭr
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Nonempty Subset ◽  
Dense Subset ◽  
And Topology ◽  
Scattered Spaces

AbstractCompactification is the process or result of making a topological space into a compact space. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. There are a lot of compactification methods but we study with Fan- Gottesman compactification. A topological space X is said to be scattered if every nonempty subset S of X contains at least one point which is isolated in S. Compact scattered spaces are important for analysis and topology. In this paper, we investigate the relation between the Fan-Gottesman compactification of T3 space and scattered spaces. We show under which conditions the Fan-Gottesman compactification X* is a scattered.

The Spatial Dimensions of Social Networks

2020 ◽  
pp. 367-383
Author(s):  
Zachary P. Neal
Keyword(s):  
Social Networks ◽  
Topological Space ◽  
Network Science ◽  
Spatial Dimensions

The first law of geography holds that everything is related to everything else, but near things are more related than distant things, where distance refers to topographical space. If a first law of network science exists, it would similarly hold that everything is related to everything else, but near things are more related than distant things, but where distance refers to topological space. Frequently these two laws collide, together holding that everything is related to everything else, but topographically and topologically near things are more related than topographically and topologically distant things. The focus of the spatial study of social networks lies in exploring a series of questions embedded in this combined law of geography and networks. This chapter explores the questions that have been asked and the answers that have been offered at the intersection of geography and networks.

On a topological construction of Juhasz and Shelah

1992 ◽  
Vol 57 (1) ◽  
pp. 166-171
Author(s):  
Dan Velleman
Keyword(s):  
Topological Space ◽  
Additional Structure ◽  
Homogeneous Set ◽  
Stationary Set ◽  
Topological Construction

In [2], Juhasz and Shelah use a forcing argument to show that it is consistent with GCH that there is a 0-dimensional T2 topological space X of cardinality ℵ3 such that every partition of the triples of X into countably many pieces has a nondiscrete (in the topology) homogeneous set. In this paper we will show how to construct such a space using a simplified (ω2, 1)-morass with certain additional structure added to it. The additional structure will be a slight strengthening of a built-in ◊ sequence, analogous to the strengthening of ordinary ◊k to ◊S for a stationary set S ⊆ k.Suppose 〈〈θα∣ ∝ ≤ ω2〉, 〈∝β∣α < β ≤ ω2〉〉 is a neat simplified (ω2, 1)-morass (see [3]). Let ℒ be a language with countably many symbols of all types, and suppose that for each α < ω2, α is an ℒ-structure with universe θα. The sequence 〈α∣α < ω2 is called a built-in ◊ sequence for the morass if for every ℒ-structure with universe ω3 there is some α < ω2 and some f ∈αω2 such that f(α) ≺ , where f(α) is the ℒ-structure isomorphic to α under the isomorphism f. We can strengthen this slightly by assuming that α is only defined for α ∈ S, for some stationary set S ⊆ ω2. We will then say that is a built-in ◊ sequence on levels in S if for every ℒ-structure with universe ω3 there is some α ∈ S and some f ∈ αω2 such that f(α) ≺ .

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

scholarly journals Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1781
Author(s):  
Samer Al Ghour
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Soft Sets ◽  
New Concepts ◽  
Bitopological Spaces ◽  
The Relationship ◽  
Soft Topological Space

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

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