Uniformization of topological spaces with small inductive dimension zero

1973 ◽  
Vol 205 (3) ◽  
pp. 241-242 ◽  
Author(s):  
K. A. Broughan
Keyword(s):  
Topological Spaces ◽  
Dimension Zero ◽  
Inductive Dimension ◽  
Small Inductive Dimension

scholarly journals Small and large inductive dimension for ideal topological spaces

2021 ◽  
Vol 22 (2) ◽  
pp. 417
Author(s):  
Fotini Sereti
Keyword(s):  
Topological Spaces ◽  
Covering Dimension ◽  
Basic Properties ◽  
Inductive Dimension ◽  
Initial Stage ◽  
Small Inductive Dimension ◽  
Important Field

<p>Undoubtedly, the small inductive dimension, ind, and the large inductive dimension, Ind, for topological spaces have been studied extensively, developing an important field in Topology. Many of their properties have been studied in details (see for example [1,4,5,9,10,18]). However, researches for dimensions in the field of ideal topological spaces are in an initial stage. The covering dimension, dim, is an exception of this fact, since it is a meaning of dimension, which has been studied for such spaces in [17]. In this paper, based on the notions of the small and large inductive dimension, new types of dimensions for ideal topological spaces are studied. They are called *-small and *-large inductive dimension, ideal small and ideal large inductive dimension. Basic properties of these dimensions are studied and relations between these dimensions are investigated.</p>

Small and Large Inductive Dimensions of Intuitionistic Fuzzy Topological Spaces

2020 ◽  
Vol 12 (3) ◽  
pp. 413-417
Author(s):  
Abdulgawad A. Q. Al-Qubati ◽  
M-El Sayed ◽  
Hadba F. Al-Qahtani
Keyword(s):  
Topological Property ◽  
Topological Spaces ◽  
Fundamental Properties ◽  
Intuitionistic Fuzzy ◽  
Inductive Dimension ◽  
Small Inductive Dimension ◽  
Fuzzy Topological Spaces

The main purpose of this paper is to present some fundamental properties of small and large inductive dimensions in intuitionistic fuzzy topological spaces. Our results can be regarded as a study of their properties such as proves subset theorems, zero dimensionality and topological property of an intuitionistic fuzzy small inductive dimension. Furthermore, we introduce a large inductive dimension of intuitionistic fuzzy bi-compact and normal spaces.

Shape morphisms and components of movable compacta

1988 ◽  
Vol 103 (3) ◽  
pp. 481-486
Author(s):  
José M. R. Sanjurjo
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
The Other ◽  
Compact Metric Space ◽  
Dimension Zero ◽  
Compact Metric Spaces ◽  
Inductive Dimension ◽  
Other Hand ◽  
Small Inductive Dimension ◽  
The Relationship

The relationship between components and movability for compacta (i.e. compact metric spaces) was described by Borsuk in [5]. Borsuk proved that if each component of a compactum X is movable, then so is X. More recently Segal and Spiez[19], motivated by results of Alonso Morón[1], have constructed a (non-compact) metric space X of small inductive dimension zero and such that X is non-movable. The construction of Segal and Spiez was based on the famous space of P. Roy [16]. On the other hand, K. Borsuk gave in [5] an example of a movable compactum with non-movable components. The structure of such compacta was studied by Oledzki in [15], where he obtained an interesting result stating that if X is a movable compactum then the set of movable components of X is dense in the space of components of X. Oledzki's result was later strengthened by Nowak[14], who proved that if all movable components of a movable compactum X are of deformation dimension at most n, then so are the non-movable components and the compactum X itself.

scholarly journals Small inductive dimension and Alexandroff topological spaces

2014 ◽  
Vol 168 ◽  
pp. 103-119 ◽  
Author(s):  
Dimitris N. Georgiou ◽  
Athanasios C. Megaritis ◽  
Seithuti P. Moshokoa
Keyword(s):  
Topological Spaces ◽  
Inductive Dimension ◽  
Small Inductive Dimension

scholarly journals Small Inductive Dimension of Topological Spaces. Part II

2009 ◽  
Vol 17 (3) ◽  
pp. 219-222
Author(s):  
Karol Pąk
Keyword(s):  
Topological Spaces ◽  
Basic Properties ◽  
Inductive Dimension ◽  
Small Inductive Dimension

Small Inductive Dimension of Topological Spaces. Part II In this paper we present basic properties of n-dimensional topological spaces according to the book [10]. In the article the formalization of Section 1.5 is completed.

scholarly journals Dimension theory and fuzzy topological spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-8
Author(s):  
S. S. Benchalli ◽  
B. M. Ittanagi ◽  
P. G. Patil
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Dimension Theory ◽  
Dimension Function ◽  
Inductive Dimension ◽  
Fuzzy Topological Space ◽  
Small Inductive Dimension ◽  
Dimension Functions ◽  
Fuzzy Topological Spaces

J. M. Aarts introduced and studied a new dimension function,Hind, in 1975 and obtained several results on this function. In this paper, a new local inductive dimension function called local huge inductive dimension function denoted bylocHindis introduced and studied. Furthermore, an effort is made to introduce and study dimension functions for fuzzy topological spaces. It has been possible to introduce and study the small inductive dimension functionindfXand large inductive dimension functionindfXfor a fuzzy topological spaceX.

Efficient Computation of the Large Inductive Dimension Using Order- and Graph-theoretic Means

2020 ◽  
Vol 177 (2) ◽  
pp. 95-113
Author(s):  
Rudolf Berghammer ◽  
Henning Schnoor ◽  
Michael Winter
Keyword(s):  
Polynomial Algorithm ◽  
Topological Spaces ◽  
Specific Class ◽  
Efficient Computation ◽  
Graph Theoretic ◽  
Inductive Dimension ◽  
Maximal Height ◽  
Analysis And Synthesis ◽  
Small Inductive Dimension ◽  
Finite Topological Spaces

Finite topological spaces and their dimensions have many applications in computer science, e.g., in digital topology, computer graphics and the analysis and synthesis of digital images. Georgiou et. al. [11] provided a polynomial algorithm for computing the covering dimension dim(X; 𝒯) of a finite topological space (X; 𝒯). In addition, they asked whether algorithms of the same complexity for computing the small inductive dimension ind(X; 𝒯) and the large inductive dimension Ind(X; 𝒯) can be developed. The first problem was solved in a previous paper [4]. Using results of the that paper, we also solve the second problem in this paper. We present a polynomial algorithm for Ind(X; 𝒯), so that there are now efficient algorithms for the three most important notions of a dimension in topology. Our solution reduces the computation of Ind(X; 𝒯), where the specialisation pre-order of (X; 𝒯) is taken as input, to the computation of the maximal height of a specific class of directed binary trees within the partially ordered set. For the latter an efficient algorithm is presented that is based on order- and graph-theoretic ideas. Also refinements and variants of the algorithm are discussed.

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