Moore spaces and uniform spaces

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.

Collectionwise Hausdorff versus collectionwise normal with respect to compact sets

Parametrizing open universals

PARACOMPACT SUBSETS

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,.

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.

d-20 - H-Closed 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.

We formulate an axiom, HYP, and from it construct a normal, metacompact, nonmetrizable Moore space. HYP unifies two better known axioms. The Continuum Hypothesis implies HYP; the nonexistence of an inner model with a measurable cardinal implies HYP. As a consequence, it is impossible to replace Nyikos’ "provisional" solution to the normal Moore space problem with a solution not involving large cardinals. Finally, we discuss how this space continues a process of lowering the character for normal, not collectionwise normal spaces.

In this paper, we introduce the concept of soft g*closed sets and soft g* open sets in soft topological space which are defined over an initial universe set with a fixed set of parameters together with its corresponding soft g*closure and soft g*interior operators. Also, we introduce the concept of soft and regular spaces and investigate the relationship between them.

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.

We consider some questions concerning the nature and size of chains of open sets in Hausdorff spaces. The following results are obtained. Theorem 1. For every cardinal κ \kappa there exists a space X X in which all discrete subsets have cardinality at most κ \kappa and which contains a chain of ( 2 κ ) + {({2^\kappa })^ + } open sets. Theorem 2. If X X is regular and contains a chain of ( 2 κ ) + {({2^\kappa })^ + } open sets, then X × X X \times X contains a discrete subset of cardinality κ + {\kappa ^ + } . Theorem 3. Let M ( X ) M(X) denote the set of all maximal chains of open subsets of X X endowed with the Tychonoff topology. (i) | M ( X ) | ⩽ 2 w ( X ) \left | {M(X)} \right | \leqslant {2^{{\text {w}}(X)}} , and (ii) ψ ( M ( X ) ) ⩽ w ( X ) \psi (M(X)) \leqslant {\text {w}}(X) . Here w ( X ) {\text {w}}(X) denotes the weight of the space X X and ψ ( M ( X ) ) \psi (M(X)) denotes the pseudocharacter of the space M ( X ) M(X) .

k-10 - Manifold

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

c-6 - Extensions of Maps