Fibrewise Fuzzy Separation Axioms

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

A categorical characterization of a subcategory S of TOP (or 7J) is one which enables the identification of S in TOP (or 7^) without requiring the reconstruction of the topological structure of its objects.In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory. Introduction. A categorical characterization of a subcategory S of TOP (or of T2) is one which enables the identification of S in TOP (or T2) without requiring the reconstruction of the topological structure of its objects.The proof of the triviality of the automorphism class group of TOP [Fe, p. 32] assures us of the existence of such characterizations.As will be shown in §1, this proof establishes that all the topological structure of an object of TOP can be recovered from the behavior of the arrows with that object as source and/ or target.In §2 we establish a number of lemmas (mostly known), and we use these lemmas in §3 to characterize many important subcategories of TOP and/ or T2, such as the metrizable spaces, the paracompact Hausdorff spaces, the normal spaces, etc.We can distinguish at least three classes of "categorical" characterizations of subcategories of TOP, which are listed below: Class 3. Straightforward translations of the familiar definitions.These can always be obtained because of the results in §1.Example: Characterize the concepts "open", "cover", "refinement", "locally finite" categorically and then the standard definition of paracompact spaces is categorical.Class 2. Characterizations using standard categorical constructions (product, coproduct, epimorphic images, etc.) but in terms of individual objects, and/or constants outside the subcategory.Example: The category of paracompact Hausdorff spaces contains precisely those Hausdorff spaces X such that

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

The present paper studies certain low-level separation axioms of a topological space, denoted by A(X), induced by a geometric AC-complex X. After proving that whereas A(X) is an Alexandroff space satisfying the semi-T1 2 -separation axiom, we observe that it does neither satisfy the pre T1 2 -separation axiom nor is a Hausdorff space. These are main motivations of the present work. Although not every A(X) is a semi-T1 space, after proceeding with an edge to edge tiling (or a face to face crystallization) of Rn, n ? N, denoted by T(Rn) as an AC complex, we prove that A(T(Rn)) is a semi-T1 space. Furthermore, we prove that A(En), induced by an nD Cartesian AC complex Cn = (En,N,dim), is also a semi-T1 space, n ? N. The paper deals with AC-complexes with the locally finite (LF-, for brevity) property, which can be used in the fields of pure and applied mathematics as well as digital topology, computational topology, and digital geometry.

A Hausdorff space X is said to be paracompact [2] provided that if W is a collection of open sets covering X, there exists a collection W' of open sets covering X such that (1) each element of W' is a subset of an element of W and (2) each point of X belongs to an open set which intersects only a finite number of elements of W'; X is said to be countably paracompact [3] provided that if W is a countable collection of open sets covering X, there exists a collection W' satisfying the above conditions. It is known that not every normal Hausdorff space is paracompact [2], but the question whether every such space is countably paracompact is as yet unsolved (cf. [3]). Since every linearly ordered space' is a normal Hausdorff space (cf. [1, p. 39]) but not necessarily paracompact [2], it seems natural to inquire whether every linearly ordered space must be countably paracompact. The purpose of the present note is to show that this is the case. DEFINITIONS. 1. A collection G of subsets of a space X is said to be locallyfinite provided every point of X belongs to an open set X which intersects at most a finite number of the elements of G. 2. If G and H are collections of sets, then H is said to be a refinement of G provided each element of H is a subset of some element of G. 3. A collection G of sets is said to be coherent provided G is not the sum of two collections G, and G2 such that no element of G1 intersects an element of G2. 4. If p belongs to some element of the collection G of sets, then the star of p with respect to G is the sum of the elements of G which contain p. Suppose X is a linearly ordered space and W is a countable collection of open sets covering X. For each point p of X, let Mp denote the set of all points x of X such that there is a finite, coherent collection HX of open intervals of X such that H. is a refinement of W and covers the set whose elements are p and x. For each point p of X, let Kp denote the collection of all open intervals k of X such that k is a subset of Mp and of some element of W. It is easily seen that for each p in X, Mp is both open and closed and if q M, then M = M,.

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.

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.

d-20 - H-Closed Spaces

c-6 - Extensions of Maps

Comparing Cartesian closed categories of (core) compactly generated spaces

In this chapter, we first define continuous mappings and their various equivalent formulations. We then consider various separation axioms for topological spaces. We prove the existence (Urysohn’s lemma) of continuous functions separating disjoint closed subsets of a normal space. Here and later we sometimes will denote a topological space E with a topology T = {U} by E u .

k-10 - Manifold

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.

This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved under uniform assumptions and with partly simpler proofs, in particular, the theorem that every computably regular computable topological space with non-empty base elements can be embedded into a computable metric space. Most of the computable separation axioms remain true for finite products of spaces.