Topology

2021 ◽  
pp. 126-144
Author(s):  
James Davidson
Keyword(s):  
Topological Spaces ◽  
Product Spaces ◽  
Product Topology ◽  
Weak Topologies ◽  
Tychonoff Theorem

This chapter discusses topological spaces and associated concepts, including first‐ and second‐countability, compactness, and separation properties. Weak topologies are defined. Product spaces, the product topology, and the Tychonoff theorem are treated and also ideas of embedding, compactification, and metrization.

scholarly journals Some Remarks about Product Spaces

2018 ◽  
Vol 26 (3) ◽  
pp. 209-222
Author(s):  
Sebastian Koch
Keyword(s):  
Product Space ◽  
Topological Spaces ◽  
Not Given ◽  
Product Spaces ◽  
Product Topology ◽  
Technical Aspects

Summary This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like [7] and [6], not even by Bourbaki in [4]. Let {Ti}i∈I be a family of topological spaces. The prebasis of the product space T = ∏i∈I Ti is defined in [5] as the set of all π−1i(V) with i ∈ I and V open in Ti. Here it is shown that the basis generated by this prebasis consists exactly of the sets ∏i∈I Vi with Vi open in Ti and for all but finitely many i ∈ I holds Vi = Ti. Given I = {a} we have T ≅ Ta, given I = {a, b} with a≠ b we have T ≅ Ta ×Tb. Given another family of topological spaces {Si}i∈I such that Si ≅ Ti for all i ∈ I, we have S = ∏i∈I Si ≅ T. If instead Si is a subspace of Ti for each i ∈ I, then S is a subspace of T. These results are obvious for mathematicians, but formally proven here by means of the Mizar system [3], [2].

scholarly journals A Tychonoff theorem in intuitionistic fuzzy topological spaces

2004 ◽  
Vol 2004 (70) ◽  
pp. 3829-3837
Author(s):  
Doğan Çoker ◽  
A. Haydar Eş ◽  
Necla Turanli
Keyword(s):  
Topological Space ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Topological Spaces ◽  
Fuzzy Topology ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Tychonoff Theorem

The purpose of this paper is to prove a Tychonoff theorem in the so-called “intuitionistic fuzzy topological spaces.” After giving the fundamental definitions, such as the definitions of intuitionistic fuzzy set, intuitionistic fuzzy topology, intuitionistic fuzzy topological space, fuzzy continuity, fuzzy compactness, and fuzzy dicompactness, we obtain several preservation properties and some characterizations concerning fuzzy compactness. Lastly we give a Tychonoff-like theorem.

scholarly journals Topological Games on Product Spaces and its Fuzzification

2013 ◽  
Vol 2 ◽  
pp. 11-15
Author(s):  
Bidyanand Prasad ◽  
BP Kumar
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Perfect Information ◽  
Topological Spaces ◽  
Product Spaces ◽  
Pursuit And Evasion

This paper is concerned with the introduction of an infinite positional game of pursuit and evasion over an ideal of a topological space. A topological game has been played over some new D-product and C-product spaces of two Hausdorff topological spaces. Perfect information, decisions and goals in a game may not be feasible. Hence, fuzzy set theory has been applied in this paper to obtain better results. Academic Voices, Vol. 2, No. 1, 2012, Pages 11-15 DOI: http://dx.doi.org/10.3126/av.v2i1.8278

scholarly journals Fuzzy Soft Compact Topological Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Seema Mishra ◽  
Rekha Srivastava
Keyword(s):  
Topological Spaces ◽  
Fuzzy Topological Spaces ◽  
Tychonoff Theorem

In this paper, we have studied compactness in fuzzy soft topological spaces which is a generalization of the corresponding concept by R. Lowen in the case of fuzzy topological spaces. Several basic desirable results have been established. In particular, we have proved the counterparts of Alexander’s subbase lemma and Tychonoff theorem for fuzzy soft topological spaces.

scholarly journals New Results on the Aggregation of Norms

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2291
Author(s):  
Tatiana Pedraza ◽  
Jesús Rodríguez-López
Keyword(s):  
Vector Space ◽  
Metric Spaces ◽  
Cartesian Product ◽  
Topological Spaces ◽  
Factor Space ◽  
Natural Question ◽  
Product Topology ◽  
Aggregation Procedure ◽  
Arbitrary Collection

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

scholarly journals Two preservation results for countable products of sequential spaces

2007 ◽  
Vol 17 (1) ◽  
pp. 161-172 ◽  
Author(s):  
MATTHIAS SCHRÖDER ◽  
ALEX SIMPSON
Keyword(s):  
Topological Spaces ◽  
Product Spaces ◽  
Convergence Spaces ◽  
Countable Product ◽  
Discrete Spaces ◽  
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.

Some Classes of θ-Compactness

1986 ◽  
Vol 29 (1) ◽  
pp. 54-59
Author(s):  
S. Broverman
Keyword(s):  
Closed Subset ◽  
Closed Subspace ◽  
Discrete Space ◽  
Order Topology ◽  
Topological Spaces ◽  
Topological Product ◽  
Product Spaces ◽  
Bounded Space

AbstractLet A and A denote the classes of ordinal spaces with the order topology and Σ-product spaces of the two point discrete space respectively. Characterizations are given in terms of ultrafiIters of clopen sets of those O-dimensional Hausdorff topological spaces that can be embedded homeomorphically as a closed subspace of a topological product of either spaces from the class Λ or the class Δ. Both classes consist of spaces that are ω0-bounded. An example is given of a O-dimensional Hausdorff ω0-bounded space that cannot be homeomorphically embedded as a closed subset of a product of spaces from either Λ or Δ, answering a question of R. G. Woods.

“Topologically Indexed Function Spaces and Adjoint Functors”

1982 ◽  
Vol 25 (2) ◽  
pp. 169-178
Author(s):  
S. B. Niefield
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Topological Spaces ◽  
Product Topology ◽  
Adjoint Functors ◽  
Point Space ◽  
Continuous Maps ◽  
Necessary And Sufficient

AbstractLet Top denote the category of topological spaces and continuous maps. In this paper we discuss families of function spaces indexed by the elements of a topological space T, and their relationship to the characterization of right adjoints Top/S → Top/T, where S is also a topological space. After reducing the problem to the case where S is a one-point space, we describe a class of right adjoints Top → Top/T, and then show that every right adjoint Top → Top/T is isomorphic to one of this form. We conclude by giving necessary and sufficient conditions for a left adjoint Top/T → Top to be isomorphic to one of the form − XTY, where Y is a space over T, and xT denotes the fiber product with the product topology.

scholarly journals Nearly Soft Menger Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
Tareq M. Al-shami ◽  
Ljubiša D. R. Kočinac
Keyword(s):  
Weak Type ◽  
Topological Spaces ◽  
Product Spaces ◽  
Topological Properties ◽  
Menger Spaces ◽  
Regular Open

In this paper, we define a weak type of soft Menger spaces, namely, nearly soft Menger spaces. We give their complete description using soft s-regular open covers and prove that they coincide with soft Menger spaces in the class of soft regular⋆ spaces. Also, we study the role of enriched and soft regular spaces in preserving nearly soft Mengerness between soft topological spaces and their parametric topological spaces. Finally, we establish some properties of nearly soft Menger spaces with respect to hereditary and topological properties and product spaces.

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