Some Maximal Topologies which are QHC

d-20 - H-Closed Spaces

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

In this paper we formulate a new structure of topological spaces which we call it lower normal spaces. This concept of spaces is directly and nature connection with the lower transversal continuous mappings on topological spaces. In this sense, we shall study spaces in which it is possible in the same way to separate two disjoint closed sets by a lower continuous real valued function. Applications in nonlinear functional analysis are considered. The concept of lower normal spaces is closely connected with the concept of normal topological spaces and the results of Alexandroff, Urysohn, Tietze, Lebesgue, Dieudonné, Tychonoff, Lefschetz, and Vietoris.

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a B r space.This paper is devoted to the study of sums and products of B r spaces in the category of topological spaces.We prove that, in general, sums and products of even two B r spaces need no longer be B r .On the other hand, for any B r space E the sum E θ E is again a B r space.Moreover, since Cech complete spaces are known to be B r , we ask whether a sum E θ F is B r provided that E is Cech complete and F is B r .It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space.Introduction.The notion of B r spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see [HuJ) and topological groups (see [Hu 2 ]).Originally, it goes back to V. Ptak's open mapping theorem (see [Kό], p. 35 ff).A Hausdorff topological space E is called a B r space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open.For a survey of the classical theory of B r spaces we refer the reader to Kόthe's book [Kό], where the locally convex case is treated.The linear topological case is investigated in the lecture notes [AEK].B r groups are considered by several authors.See for instance [Hu 2 ], [Ba], [Gr], [Su].In a purely topological context, B r spaces have first been investigated by Weston in [We], although the term B r space is not used there.Translated into the B r terminology, Weston proved that every completely metrizable topological space is a B r space.His result has been generalized by Byczkowski and Pol in [BP], who proved that every Cech complete topological space is a B r space.In [No] we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a B r space.We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf.[Wi]).In the present paper we examine the invariance of the class of B r spaces under topological sums and products.It turns out that, in general, the sum of even two B r spaces need not be a B r space.A counterexample is given in §2.In §1 we obtain a positive result stating that the sum E θ E

The present paper comes across the main steps that were laid from Zadeh’s fuzziness and Atanassov’s intuitionistic fuzzy sets to Smarandache’s indeterminacy and to Molodstov’s soft sets. Two hybrid methods for assessment and decision making, respectively, under fuzzy conditions are also presented using suitable examples that use soft sets and real intervals as tools. The decision making method improves on an earlier method of Maji et al. Further, it is described how the concept of topological space, the most general category of mathematical spaces, can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kinds of structures. In particular, fuzzy and soft topological spaces are defined and examples are given to illustrate these generalizations.

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

In a topological spaceX, a T2-distinct pointx means that for anyy∈X x≠y, there exist disjoint open neighbourhoods ofx andy. Similarly, T0-distinct points and T1distinct points are defined. In a Ti-distinct point-setA, we assume that eachx∈A is a Ti-distinct point (i=0, 1, 2). In the present paper some implications of these notions which ‘localize’ the Ti-separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T2-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T0-distinct points are used to give two characterizations of the RD-axiom.1 In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an ω-limit point is stated.

A note on [formula omitted]-spaces and related conclusions

We build a topological model, based on intuitionistic logic, for multi-agent biological systems (such as Physarum polycephalum, bacterial colonies or any other swarm), reacting to external nourishment stimuli. Our construction follows the topological description of brain activity, where particles (neurons) are activated by an external environment, represented by a topological space X with an open cover {U_i:iin I}. The brain builds the model of this external space via the nerve (trace) of a topological space X. Here the body of Physarum polycephalum or a swarm made of networks of tubular structures represents a nerve (trace) of X also which means that Physarum polycephalum or a swarm gains orientation in the space of external stimuli even in the absence of any neural system. The logic of living organisms is based on open subsets of X and thus can be represented by Heyting algebra (i.e. intuitionistically). We also consider the generalisation of the nerve construction to a categorical context, where the category is determined by the network structures of multi-agent biological system. This model can be generalised up to simulating the behaviour of any swarm by means of intuitionistic logic.

It is shown that there is an integer v(n) < 32n+1 such that any n-dimensional paracompact space X has intersection multiplicity at most v(n). That is, if Ii is an open cover of X, then there is an open cover (9 refining iJ such that any element of t9 intersects at most v(n) elements of O. For any open cover CU of a topological space X define m('U) to be the maximum number of elements of cU that any member of cU can intersect. X is said to have intersection multiplicity at most m if every open cover CU of X has an open refinement (9 covering X such that m(Q) < m. The intersection multiplicity of X is then the least integer m such that X has intersection multiplicity at most m and is denoted m(X). Intersection multiplicity is clearly preserved by homeomorphisms. Also if A is a closed subset of X, then m(A) < m(X)X Received by the editors February 15, 1974 and, in revised form, May 7, 1974. AMS (MOS) subject classifications (1970). Primary 57C05, 54D20, 54F45, 55C10.

Analysis and Algebra on Differentiable Manifolds

Functorial admissible quasi-uniformities on topological spaces

In the theory of frames (or "pointless topologies"), several authors have tried to find a suitable form of separation axioms.Our purpose is to describe a T2-axiom in the form usual in the case of regular frames.T2-frames coincide for topological spaces with Hausdorff spaces but they are described independently on points.We also investigate almost compact frames and H-closed extensions of T2-frames (see [3], 6.1.,h, for spaces).All unexplained facts concerning frames can be found in Johnstone [10] or in [16].Recall that a frame is a complete lattice L in which the infinite distributive law aA\/S=\/{aAs: s G 5} holds for all a € L, S C L.The known facts (see [10] or [16]) indicate the importance of the opposite category Loc of locales to the category Frm of frames.We will work in the category Frm.If Top is the category of topological spaces, then the set 0(T) of all open sets of T E Top is a frame.Frames isomorphic to some 0(T) are called spatial (or a topology).In [16], frames in which primes (i.e.A-irreducible and -j--1 elements) are dual atoms (called T\-frames), are investigated.The category of all T\-frames is the smallest epireflective subcategory in Frm containing all T\-spaces.Every T\-frame is a homomorphic image of a Ti-space and every spatial Ti-frame is a Ti-space.We will now investigate similar problems for T2-fra1r.es.The category of T^-frames is an epireflective subcategory in Frm containing all Hausdorff spaces but there exists a To-frame which is not a homomorphic image of a Hausdorff topology.Spatial To-frames are Hausdorff spaces.There exists a compact To-frame which is not regular.An almost compact frame which is a homomorphic image of a HausdorfT topology is a topology.A compact T2-frame exists which is not