Hausdorff Separation Research Articles

Linear pointwise L-quasi uniformities

The main purpose of this paper is to study pointwise L-quasi uniformities on vector spaces. Firstly, we introduced a concept of linear pointwise L-quasi uniformities and proved that there is a one-to-one correspondence between the family of linear pointwise L-quasi uniformities and the set of L-topological vector spaces. More precisely, each linear pointwise L-quasi uniformity can generate a vector L-topology and every L-topological vector space is linear pointwise L-quasi uniformizable. Secondly, the Hausdorff separation and pointwise quasi-uniformly continuous of linear operators are investigated. In addition, as an example, the linear pointwise L-quasi uniformity induced by an L-fuzzy norm is also discussed.

Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces

In this paper, we characterize each of T0, T1, Pre-Hausdorff and Hausdorff separation properties for the category L-GS of quantale-valued gauge spaces and L-gauge morphisms. Moreover, we investigate how these concepts are related to each other in this category. We show that T0, T1 and T2 are equivalent in the realm of Pre-Hausdorff quantale-valued gauge spaces. Finally, we compare our results with the ones in some other categories.

The topology on the space of measurable functions that is compatible with convergence in nonadditive measure

In this paper, given a nonadditive measure μ, a topology, which is compatible with convergence in μ-measure, is defined on the real linear space of all (equivalence classes of) measurable functions by using the distance introduced by Dunford and Schwartz. Some fundamental properties of the topology, such as the open sphere condition, the Hausdorff separation axiom, separability, approximation by continuous functions, linearlization, and metrizability, are related to the characteristics of a nonadditive measure.

Linear fuzzifying uniformities

The main purpose of this paper is to study Hutton type fuzzifying uniformities on linear spaces. Firstly, we show that if a base of a fuzzifying uniformity defined over a linear space is translation-invariant, balanced and absorbed, then it generates a linear fuzzifying topology. From this linear fuzzifying topology, we can construct a new linear fuzzifying uniformity (i.e., a fuzzifying uniformity compatible with the linear structure) which is equivalent to the original fuzzifying uniformity. Secondly, the Hausdorff separation and complete boundedness in linear fuzzifying uniformities are investigated. In addition, as an example, the linear fuzzifying uniformity induced by a fuzzy norm is also discussed.

Compactly Generated Spaces and Quasi-spaces in Topology

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $$\textsf {Top}$$ of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class $$\mathcal {C}$$ of objects we generalize the notion of $$\mathcal {C}$$-generated spaces, from which we derive, for instance, a general concept of Alexandroff spaces. Furthermore, as done for $$\textsf {Top}$$, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

Fuzzifying topologies on the space of linear operators

We introduce the notion of fuzzifying topologies on the space of linear operators. The concepts of fuzzifying topologies of pointwise convergence, bounded convergence, and complete bounded convergence are presented. We prove a sufficient and necessary condition for which the fuzzifying topologies are compatible with the structure on the space of linear operators. The Hausdorff separation of a fuzzifying topology on the space of linear operators is discussed. Specially, the net convergence, fuzzy boundedness, and fuzzy continuity of evaluation mappings in the fuzzifying topological linear space of pointwise convergence are investigated.

Categorical foundations of topology with applications to quantaloid enriched topological spaces

Based on ordered monads this paper uncovers the categorical basis of topology in terms of a categorical formulation of neighborhood axioms. Here dense subobjects, lower separation axioms and regularity receive a purely categorical representation. In the case of appropriate submonads of the double presheaf monad this theory is applied to quantaloid-enriched topological spaces which form a common framework for many valued topology as well as for non-commutative topology. As an illumination of this situation two examples are given: the first one is chosen from probability theory and has the following characteristics: Weak convergence of τ-smooth probability measures is topological. The Hausdorff separation axiom is valid. Dirac measures form a dense subset. The second example is related to operator theory and explains the topologization of spectra of non-commutative C*-algebras.

Weak Hausdorff separation axiom in I-fuzzy topological spaces and its application

In this paper, the weak Hausdorff separation axiom in the sense of Fang and Ren [A set of new separations axioms in L-fuzzy topological spaces, Fuzzy Sets and Systems 96 (1998) 359–376] is generalized from L-topologies to I-fuzzy topologies. It is shown that this notion satisfies the hereditary, productive properties, and the relations between it and other separation axioms are obtained. Also the degree to which an I-fuzzy topological space is weak Hausdorff is studied in terms of the fuzzy nets and the corresponding level I-topologies. As an application, we prove that the degree to which an I-fuzzy topological group is weak Hausdorff is equal to the degree to which it is T 1.

Generalized locally bounded L-topological vector spaces

In this paper, the notion of generalized locally bounded L-topological vector spaces is proposed. The relationship between this kind of space and locally bounded L-topological vector space as defined by Yan and Fang [Locally bounded L-topological vector spaces, Inf. Sci. 159 (2004) 273–281] is investigated. In addition, the concept of a family of generalized L-fuzzy quasi-norms is introduced. Based on this notion, generalized locally bounded L-topological vector spaces are characterized. Finally, the Hausdorff separation property, convergence of molecule nets and boundedness of L-fuzzy sets in generalized locally bounded L-topological vector spaces are studied.

L-topological vector spaces and families of L-fuzzy pseudo-norms

In this paper, the concept of a family of L-fuzzy pseudo-norms on vector spaces is proposed and the characterization of L-vector topologies in terms of a family of L-fuzzy pseudo-norms is presented. As applications of the characterization, the Hausdorff separation property, convergence of molecule nets and boundedness of L-sets in L-topological vector spaces are investigated.

New definition of locally convex L-topological vector spaces

In this paper, a new definition of locally convex L-topological vector spaces is given. The relationship between this new definition and the previous definition of locally convex L-topological vector spaces given by Yan and Fang in 1999 is investigated. Moreover, the concept of generalized L-fuzzy semi-norm is introduced. By using a family of generalized L-fuzzy semi-norms, a characterization of the new locally convex L-topological vector spaces is presented. Finally, as applications of this characterization, the Hausdorff separation property, convergence of molecule nets and boundedness of L-fuzzy sets in locally convex L-topological vector spaces are studied.

A note on Hausdorff separation in L-TOP

This note shows that a careful study of the history of Hausdorff-type separation axioms in many valued topology is necessary. We make a few observations regarding a Hausdorff-type axiom for L-topological spaces reinvented recently by Shi. In particular, we put it into the adjoint situation associated with the functors ω L and ι L with L a continuous lattice.

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

In a topological construct $\mathcal{X}$ endowed with a proper $(\mathcal{E}, \mathcal{M})$ -factorization system and a concrete functor $\Lambda:\mathcal{X}\rightarrow \mathbf{Prap}$ , we study $\mathcal{F}_{\Lambda}$ -compactness and $\mathcal{F}_{\Lambda}$ -Hausdorff separation, where $\mathcal{F}_{\Lambda}$ is a class of “closed morphisms” in the sense of Clementino et al. (A functional approach to general topology. In: Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge University Press, Cambridge, 2004), determined by Λ. In particular, we point out under which conditions on Λ, the notion of $\mathcal{F}_{\Lambda}$ -compactness of an object $\underline{X}$ of $\mathcal{X}$ coincides with 0-compactness of the image $\Lambda(\underline{X})$ in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b-compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of b-compactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability.

A characterization of Hausdorff separation for a special class of varieties

In 1997, Coleman suggested that congruence modularity and n-permutability are necessary and sufficient properties for a variety to have all of its T0-topological algebras be Hausdorff. The sufficiency part of this statement was later shown by Kearnes and Sequeira. Here we show that necessity holds for a certain subclass of varieties, hence providing a characterization of the property $$ T_{0} \Rightarrow T_{2} $$ in this particular case.

Extremal pseudocompact topological groups

Topological groups here are assumed to satisfy the Hausdorff separation property. A topological group G is totally bounded if it embeds as a (dense) subgroup of a compact group G ¯ ; here G ¯ , the Weil completion of G, is unique in the obvious sense. It is known that every pseudocompact topological group is totally bounded; and a totally bounded group G is pseudocompact if and only if G meets each nonempty G δ -subset of G ¯ . A pseudocompact group is said to be r- extremal [resp., s- extremal] if G admits no strictly finer pseudocompact group topology [resp., G has no proper dense pseudocompact subgroup]. (Note: r- derives from “refinement'', s- from “subgroup''.) Let P be the class of non-metrizable, pseudocompact Abelian groups. The authors contribute to the growing literature (see for example J. Galindo, Sci. Math. Japonicae 55 (2001) 627) supporting the conjecture that no G ∈ P is either r- nor s-extremal—but that conjecture remains open. Except for portions of (a), the following are new results concerning G ∈ P proved here. The proofs derive largely from basic, sometimes subtle, considerations comparing the algebraic structure of G ∈ P with the algebraic structure of the Weil completion G ¯ . (a) If G is either r- or s-extremal, then r 0 ( G ) = c < w ( G ) . (b) If G is totally disconnected, then G is neither r- nor s-extremal. (c) If G is either torsion-free or countably compact, then G is not both r- and s-extremal. (d) Not every closed, pseudocompact subgroup N of G is s-extremal; if G itself is either r- or s-extremal then the witnessing N may be chosen connected. (e) If in some G ∈ P every closed subgroup in P is r-extremal, then there is connected H ∈ P with the same property. (f) If 2 ω = 2 ω 1 and every closed subgroup of G is pseudocompact, then G is not s-extremal.

Metrization theorems in L-topological spaces

In this paper, some characterizations of a pointwise uniformity are presented in L-fuzzy set theory. A new Hausdorff separation axiom compatible with pointwise metric is introduced. Urysohn metrization theorem and Alexandroff–Urysohn metrization theorem in general topology are generalized to L-topology.

Feuilletages transversalement projectifs sur les variétés de Seifert

Let M be a Seifert manifold with non-solvable fundamental group. Let Φ be a one- dimensional foliation on M, equipped with a transverse real projective structure. We assume moreover that Φ satisfies the Homotopy Lifting Property, i.e., that the leaf space of the lifting of Φ in the universal covering of M satisfies the Hausdorff separation property. Then, up to finite coverings, Φ belongs to one of the following three families of transversely projective foliations: the family of projective fibrations, the family of convex geodesic foliations, or the family of projective horocyclic foliations.

Separatedness in constructive topology

We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework.

A note on the comparison of topologies

A considerable problem of some bitopological covering properties is the bitopological unstability with respect to the presence of the pairwise Hausdorff separation axiom. For instance, if the space is RR‐pairwise paracompact, its two topologies will collapse and revert to the unitopological case. We introduce a new bitopological separation axiom τS2σ which is appropriate for the study of the bitopological collapse. We also show that the property that may cause the collapse is much weaker than some modifications of pairwise paracompactness and we generalize several results of T. G. Raghavan and I. L. Reilly (1977) regarding the comparison of topologies.

A note on Raghavan‐Reilly′s pairwise paracompactness

The bitopological unstability of RR‐pairwise paracompactness in presence of pairwise Hausdorff separation axiom is caused by a bitopological property which is much weaker and more local than RR‐pairwise paracompactness. We slightly generalize some Michael′s constructions and characterize RR‐pairwise paracompactness in terms of bitopological θ‐regularity, and some other weaker modifications of pairwise paracompactness.