Variations on known and recent cardinality bounds

Additivity of metrizability and related properties

PARACOMPACT SUBSETS

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.

A weak regular space called almost regular space and the strong Hausdorff space in L-fuzzy topological space are introduced. Their properties have also been investigated in This work. The strong Hausdorff spaces are proved to have interesting properties such as topological properties, product properties, hereditary properties and L-good extension. Their relationship between Hausdorff spaces and regular spaces are also studied.

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

One-point connectifications of regular spaces

k-10 - Manifold

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.

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.

The main purpose of this work is to apply elements of set theory to the study of biorthogonal systems in Banach spaces. Initially, we study Markushevic's Theorem, which ensures that every separable Banach space has a countable biorthogonal system. With this result, we focus our attention to the study of nonseparable Banach spaces, more especifically, we study the existence of uncountable biorthogonal systems in Banach spaces of the form C(K), with K a nonmetrizable compact Hausdorff space. In this direction, we study two theorems of S. Todorcevic. The first one gives us sufficient conditions that a compact Hausdorff space K must satisfy in order to get that the respective Banach space C(K) has an uncountable biorthogonal system. The second one tells us that under Martin's Axiom, every nonseparable Banach space of the form C(K) has an uncountable biorthogonal system. Next, we consider some cardinal functions defined by P. Koszmider for Banach spaces, related with biorthogonal systems, and we study its relations with well -known cardinal functions. In particular, we obtain an original result relating the weight of a compact Hausdorff space K to the size of certain biorthogonal systems in C(K), generalizing a result of S.

d-20 - H-Closed Spaces

c-6 - Extensions of Maps

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.

Suppose $F$ is a totally ordered field equipped with its order topology and $X$ a completely $F$-regular topological space. Suppose $\mathcal{P}$ is an ideal of closed sets in $X$ and $X$ is locally-$\mathcal{P}$. Let $C_\mathcal{P}(X, F)=\{f\colon X\rightarrow F~|~ f$ is continuous on $X$ and its support belongs to $\mathcal{P}\}$ and $C_\infty^\mathcal{P}(X, F)=\{f\in C_\mathcal{P}(X, F)~|~ \forall \varepsilon>0$ in $F$, $cl_X\{x\in X: |f(x)|>\varepsilon\}\in \mathcal{P}\}$. Then $C_\mathcal{P}(X, F)$ is a Noetherian ring if and only if $C_\infty^\mathcal{P}(X,F)$ is a Noetherian ring if and only if $X$ is a finite set. The fact that a locally compact Hausdorff space $X$ is finite if and only if the ring $C_K(X)$ is Noetherian if and only if the ring $C_\infty(X)$ is Noetherian, follows as a particular case on choosing $F=\mathbb{R}$ and $\mathcal{P}=$ the ideal of all compact sets in $X$. On the other hand if one takes $F=\mathbb{R}$ and $\mathcal{P}=$ the ideal of closed relatively pseudocompact subsets of $X$, then it follows that a locally pseudocompact space $X$ is finite if and only if the ring $C_\psi(X)$ of all real valued continuous functions on $X$ with pseudocompact support is Noetherian if and only if the ring $C_\infty^\psi(X)=\{f\in C(X)~|~ \forall\varepsilon>0, cl_X\{x\in X: |f(x)|>\varepsilon\}$ is pseudocompact $\}$ is Noetherian. Finally on choosing $F=\mathbb{R}$ and $\mathcal{P}=$ the ideal of all closed sets in $X$, it follows that: $X$ is finite if and only if the ring $C(X)$ is Noetherian if and only if the ring $C^*(X)$ is Noetherian.

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.