Algebraic Topology of Finite Topological Spaces and Applications
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The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory. This book is a revised version of the PhD Thesis of the author. All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest.
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Be the first to write a review. Add to Wishlist. Ships in 15 business days. In this chapter we present some results concerning elementary topological aspects of finite spaces. The proofs use basic elements of Algebraic Topology and have a strong combinatorial flavour. We study further homotopical properties including classical homotopy invariants and finite analogues of well-known topological constructions. In Sect. However, by Theorem 2. In this chapter we will study finite models of polyhedra in this sense and we will describe the minimal finite models of some well-known Hausdorff spaces, i.
The main results of this chapter appear in . Whitehead translated these algebraic moves into the well-known geometric moves of elementary collapses and expansions of finite simplicial complexes. The notion of collapse of finite spaces is directly connected with the concept of simplicial collapse.
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In Chap. In this chapter we will define the notion of strong collapse of simplicial complexes which leades to strong homotopy types of complexes. A method of reduction of finite spaces is a technique that allows one to reduce the number of points of a finite topological space preserving some properties of the space. The results of McCord show that each compact polyhedron K can be modeled, up to weak homotopy, by a finite space X K.
Many researchers heve contributed towards the algebraic structure of soft set theory. Acar et al. Feng et al. Shabir and Ali [ 7 ] studied soft ideals over a semigroup. Qiu-Mei et al. Gunduz Aras and Bayramov [ 9 , 10 ] introduced fuzzy soft modules and intuitionistic fuzzy soft modules and investigated some basic properties.
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Ozturk and Bayramov defined chain complexes of soft modules and their soft homology modules. Ozturk et al.
Recently, Shabir and Naz [ 11 ] initiated the study of soft topological spaces. Theoretical studies of soft topological spaces have also been by some authors in [ 12 — 17 ]. In the study [ 18 ] different soft point concepts from the studies [ 12 — 17 ] were given. In this study, soft point consepts in the study [ 18 ] are used. In the present study, first soft compact set on soft topological spaces is defined and the properties of this soft compact set are investigated. Based on this, soft compact-open topology of soft topological spaces is defined on soft mapping space.
In this section we will introduce necessary definitions and theorems for soft sets. Molodtsov [ 1 ] defined the soft set in the following way. Let X be an initial universe set and let E be a set of parameters. In other words, the soft set is a parameterized family of subsets of the set X. Let F , E be a soft set over X.
If f is soft continuous mapping for all x e , E , then f is called soft continuous mapping. When a homeomorphism f exists between X and Y , we say that X is soft homeomorphic to Y. This mapping is called an evaluation map. Now, we construct relationships between some function spaces. We define the mapping. We define mapping. We define exponential law. We define the following mapping:. From the hypothesis,. Hence from Theorem 32 , H , E is soft compact set.
Since the soft set H , E is a soft compact set, then.
The union of finite number of soft compact set on soft topological spaces is soft compact set. The generated topology from the sets. Now, let us give the soft continuous mappings set Y X. Since the soft topological space is a finite, in this space each soft set is a soft compact.
Then the soft evaluation map.
Algebraic Topology of Finite Topological Spaces and Applications | Jonathan A. Barmak | Springer
Let Y Z have the soft compact open topology. Consider the functions. Conversly suppose that f is soft continuous map. The soft mapping E is bijective.
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Hence E is soft homeomorphism. Then since. The mapping. We may take soft set U , E as. First soft compact set on soft topological spaces is defined and the properties of this soft compact set are investigated. Based on these concepts, soft compact-open topology is defined in functional spaces of soft topological spaces. Some properties of these functional spaces are studied. Proof of theorem of evaluation map, exponential law, and others are provided.
Further, these functional spaces are studied and interrelations between various functional spaces soft compact-open topology are established. To extend this work, by using soft compact-open topology on function spaces, one could study different subdisciplines of mathematics, for example, algebraic topology, functional analysis, and so forth. The authors declare that there is no conflict of interests regarding the publication of this paper.