Topology

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.

New Results on the Aggregation of Norms

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.

Another Look at Topological BCH-algebras

A BCH-algebra (H, ⁕ ,0) furnished with a topology τ on H (also called a BCH-topology on H) is called a topological BCH-algebra (or TBCH-algebra) if the function ⁕: H ×H → H, defined by ⁕((x, y)) =x ⁕ y for any x,y in H, is continuous, where the Cartesian product topology on H × H is furnished by τ. In this paper, we give other structural properties of topological BCH-algebras.

On Product of Smooth Neutrosophic Topological Spaces

In this paper, we develop the notion of the basis for a smooth neutrosophic topology in a more natural way. As a sequel, we define the notion of symmetric neutrosophic quasi-coincident neighborhood systems and prove some interesting results that fit with the classical ones, to establish the consistency of theory developed. Finally, we define and discuss the concept of product topology, in this context, using the definition of basis.

On Completeness of Sliced Spaces under the Alexandrov Topology

We show that in a sliced spacetime ( V , g ) , global hyperbolicity in V is equivalent to T A -completeness of a slice, if and only if the product topology T P , on V, is equivalent to T A , where T A denotes the usual spacetime Alexandrov “interval” topology.

Results on a Pre-T_2 Space and Pre-Stability

This paper contains an equivalent statements of a pre- space, where are considered subsets of with the product topology. An equivalence relation between the preclosed set and a pre- space, and a relation between a pre- space and the preclosed set with some conditions on a function are found. In addition, we have proved that the graph of is preclosed in if is a pre- space, where the equivalence relation on is open. On the other hand, we introduce the definition of a pre-stable ( pre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that a pre-stable ( pre-stable) set is positively invariant (invariant), and we add a condition on this set to prove the converse. Finally, a relationship between, (i) a pre-stable ( pre-stable) set and its component (ii) a pre- space and a (positively critical point) critical point, are gotten.

On Soft Rough Topology with Multi-Attribute Group Decision Making

Rough set approaches encounter uncertainty by means of boundary regions instead of membership values. In this paper, we develop the topological structure on soft rough set ( SR -set) by using pairwise SR -approximations. We define SR -open set, SR -closed sets, SR -closure, SR -interior, SR -neighborhood, SR -bases, product topology on SR -sets, continuous mapping, and compactness in soft rough topological space ( SRTS ). The developments of the theory on SR -set and SR -topology exhibit not only an important theoretical value but also represent significant applications of SR -sets. We applied an algorithm based on SR -set to multi-attribute group decision making (MAGDM) to deal with uncertainty.

Control over the macrocyclisation pathway and product topology in a copper-templated catenane synthesis

Strategies to control building block intertwining and the efficient assembly of a linear [4]catenane are presented.

Some Remarks about Product Spaces

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].