Fuzzy almost regular and fuzzy strong Hausdorff separation properties

Introduction. All spaces considered in this paper are assumed to be (Hausdorff) completely regular, and all maps are continuous. Let be a topological property of spaces. We shall identify with the class of spaces having . A space having is called a -space, and a subspace of a -space is called a -regular space. The class of -regular spaces is denoted by R(). Following [37], we call a closed hereditary, productive, topological property such that each -regular space has a -regular compactification a topological extension property, or simply, an extension property. In this paper, we restrict our attention to extension properties satisfying the following axioms:(A1) The two-point discrete space has .(A2) If each -regular space of nonmeasurable cardinal has , then = R().

In this paper, we define the notion of intuitionistic fuzzy based regular and normal spaces. We also study that classical regular and normal spaces are also intuitionistic fuzzy based regular and normal spaces but the converses are not true in general. This notion opens up a new conception of generalization of classical regular and normal spaces. The hereditary and topological properties of intuitionistic fuzzy based regular and normal spaces have been also investigated. Moreover, by setting some examples we show that every intuitionistic fuzzy based regular space as well as intuitionistic fuzzy based normal space need not be T1 spaces. Finally, it is shown that under some conditions the images and homeomorphic images are preserved in intuitionistic fuzzy based regular and normal spaces.

d-20 - H-Closed Spaces

One-point connectifications of regular spaces

Although arising from different motivations Moore spaces2 and uniform spaces [7] are generalizations of metric spaces which have considerable similarity. This can be seen from the following characterizations by means of open coverings: Let S be a regular Hausdorff space. If there exists a family { G} of open coverings of S such that (a) { G } is countable and3 (b) if U is an open set and p G U, there is an element G of { G} such that Gp*C U, then S is a Moore space and conversely [6]. If there exists a family { G} of open coverings of S such that (a') if G, and G2 are elements of { G} there is an element G3 of { G } which is a star refinement of both G, and G2 and (b) if U is an open set and p GU, there is an element G of { G such that GP*C U, then S is a uniform space4 and conversely [5]. Of course if a { G} exists so that all three conditions (a), (a') and (b) hold true simultaneously, S is metric.' Except for the obvious fact that the Hausdorff first countability axiom must hold true for Moore spaces while it need not do so for uniform spaces, it is by no means obvious that they differ otherwise. L. F. McAuley has given an example of a normal semi-metric Hausdorff space S which is not a Moore space [3]. Being normal, S is both regular and uniform and being semi-metric, the first countability axiom holds true for S. I give below an example of a locally connected, connected, complete Moore space6 containing a point p at which it is not completely regular. Hence it is not uniform [2 or 5 ]. This space is obtained by piecing together along their boundaries adjacent terms of a simple sequence

Let I_{nu }(x) and K_{nu }(x) be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by x^{alpha } Phi _{i,nu }(x), i=1,2, with Phi _{1,nu }=I_{nu -1}(x)/I_{nu }(x) and Phi _{2,nu }(x)=-K_{nu -1}(x)/K_{nu }(x), are bounds for x^{alpha } Phi _{i,nu }(x), which are solutions with unique monotonicity properties; these bounds hold at least for pm alpha notin (0,1) and nu ge 1/2. Properties for the product P_{nu }(x)=I_{nu }(x)K_{nu }(x) can be obtained as a consequence; for instance, it is shown that P_{nu }(x) is decreasing if nu ge -1 (extending the known range of this result) and that xP_{nu }(x) is increasing for nu ge 1/2. We also show that the double ratios W_{i,nu }(x)=Phi _{i,nu +1}(x)/Phi _{i,nu }(x) are monotonic and that these monotonicity properties are exclusive of the first and second kind modified Bessel functions. Sharp trigonometric bounds can be extracted from the monotonicity of the double ratios. The trigonometric bounds for the ratios and the product are very accurate as xrightarrow 0^+, xrightarrow +infty and nu rightarrow +infty in the sense that the first two terms in the power series expansions in these limits are exact.

k-10 - Manifold

In this paper, we extend the N-compactness in L-Fuzzy Topological space to intuitionistic L-Fuzzy Topological space. Firstly, intuitionistic fuzzy N-compactness is defined by α-quasi-coincident family in an intuitionistic L-Fuzzy Topological space, and its equivalent characterization is given by using intuitionistic fuzzy α-net and intuitionistic fuzzy filter. Secondly, it is proved that intuitionistic fuzzy N-compactness has closed heritability; The union of a finite number of intuitionistic fuzzy N-compact sets is still intuitionistic fuzzy N-compact set and intuitionistic fuzzy N-compactness is topological invariance. This research enriches the theoretical system of intuitionistic L-fuzzy Topological space, and provides a certain reference value for the related theoretical research of intuitionistic L-fuzzy Topological space.

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

Relations between the good definitions of compactness in L-fuzzy topological spaces

A new concept of near semicompactness is introduced in L-fuzzy topological spaces, where L is a fuzzy lattice. It is weaker than semicompactness and is stronger than S*-closedness. It is described with collection of semiclosed L-fuzzy subsets and finite p-level semiregular-semiopen cover. It is a good L-extension; it is hereditary for semiregular-semiclosed L-fuzzy subsets and semiclopen L-fuzzy subsets; it is finitely additive and invariant under almost irresolute and almost irresolute-open mappings.

On countably strong fuzzy compact sets in L-fuzzy topological spaces

Within that research, we introduce fibrewise fuzzy types of the most important separation axioms in ordinary fuzz topology, namely fibrewise fuzzy (T 0 spaces, T 1 spaces, R 0 spaces, Hausdorff spaces, functionally Hausdorff spaces, regular spaces, completely regular spaces, normal spaces, and normal spaces). Too we add numerous outcomes about it.

Additivity of metrizability and related properties

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0,1,2,3,4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0,1,4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0,1,2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1,2,3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0,1,2,3,4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.