Small and large inductive dimension for ideal topological spaces

<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>

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

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.

Small and Large Inductive Dimensions of Intuitionistic 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.

The small inductive dimension of subsets of Alexandroff spaces

We describe the small inductive dimension ind in the class of Alexandroff spaces by the use of some standard spaces. Then for ind we suggest decomposition, sum and product theorems in the class. The sum and product theorems there we prove even for the small transfinite inductive dimension trind. As an application of that, for each positive integers k,n such that k ? n we get a simple description in terms of even and odd numbers of the family S(k,n) = {S ? Kn : |S|=k+1 and indS=k}, where K is the Khalimsky line.