C-6 - Extensions of Maps

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.

PARACOMPACT SUBSETS

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.

Countable small rank and cardinal invariants II

e-2 - Classical Metrization Theorems

b-7 - Cleavable (Splittable) 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

Functorial admissible quasi-uniformities on topological spaces

The study of mappings and their properties turned out to be of central importance for mathematical analysis. Quite often it is desired that a small change in the arguments of a mapping yields only a corresponding small change in the values of the mapping. Formalizing this idea results in the notion of continuity. This chapter introduces continuous mappings in the rather general context of metric spaces. Then central properties of continuous mappings are shown and characterizations in terms of previously studied properties are proved. Subsequently, variations of continuity are considered and the problem of under what conditions a continuous mapping allows for a continuation to a larger domain is studied. We finish this chapter by looking at continuous functions over the reals. In the course of this chapter several functions considered previously will be studied again with respect to the problem of whether or not they are continuous.

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.

Real-life structures always include indeterminacy.The Mathematical tool which is well-known in dealing with indeterminacy is Neutrosophic.Smarandache proposed the approach of Neutrosophic sets.Neutrosophic sets deal with uncertain data.The notion of Neutrosophic set is generally referred to as the generalization of intuitionistic fuzzy set.In 2022, S. Pious Missier, A. Anusuya and J. Martina Jency introduced and studied the concepts of Neutrosophic generalized regular star closed eu closed *-N gr sets and Neutrosic generalized regular star eu open *-N gr sets in Neutrosophic topological spaces and studied some of their properties and characterizations as well as analyzed the relationships between these newly introduced sets and the already existing neutrosophic sets.In this paper, we introduce the notions of eu compact spaces,

The subset theorem for the covering dimension of a space X is the statement that \(\dim A \leq \dim X\) for every subspace A of X. The subset theorem for perfectly normal spaces was established by Cech. The same author in 1948 raised the question whether the subset theorem holds for hereditarily normal Hausdorff spaces (see Cech, Colloq Math 1:332, 1948). Recall that there are trivial examples of hereditarily normal (but neither T1 nor regular) spaces for which the subset theorem fails (Exercise 2.18). Also, for any \(n\in \mathbb {N}\), there is a strongly zero-dimensional normal Hausdorff space N that contains a normal space M with \(\dim M = n\) (Theorem 14.9). A negative answer to Cech’s question was first obtained by Filippov in 1973, who assumed, however, the existence of a Souslin tree. The problem was finally settled in ZFC, without additional set theoretic assumptions, by Elzbieta and Roman Pol in 1977. This chapter is devoted to presenting their example of a Hausdorff, strongly zero-dimensional, hereditarily normal space that contains a perfectly normal, locally second countable subspace of positive covering dimension.

f-10 - Topological Characterizations of Spaces

The properties and relations of three different types of intuitionistic normal spaces are studied and among that it is shown that intuitionistic strongly normal space is topologically invariant. Also weak regular space (R0 space) in intuitionistic topological space is introduced, its properties are established and its relation with other existing separation axioms are investigated.

Generalizing de Vries’ duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.