拓扑分子格的分离公理

在[1]中我们建立了拓扑分子格的理论,它既是古典的点集拓扑学的推广,又是晚近发展起来的Fuzzy拓扑学的推广,对于某些Fuzzy格L(如L是线性序集或L是分子格等),它也是L—Fuzzy拓扑学的推广。因此,凡在拓扑分子格中得到的结果自然都是上述各种拓扑学中相应定理的一般化形式。在本文中我们将讨论拓扑分子格的分离公理。 我们熟知点集拓扑学中的分离公理有多种不同的等价形式。以正则性为例,设X是拓扑空间,X叫正则的,当且仅当对每个点a∈X以及a的每个开邻域U,a有开邻域V满足条件V~-U。这一分离公理又可表述为:设a∈X,F是X中不包含a的闭集,则有开集P     

Open Ultrafilters and Separability with the Use of the Operation of Closure

We study ultrafilters of topologies as well as sets of ultrafilters that each time dominate the open neighborhood filter of some fixed point in a topological space. The sets of ultrafilters are considered as “enlarged points” of the original space. We study conditions that provide the distinguishability of (enlarged) “points” of this type. We use nontraditional separability axioms and study their connection with the known axioms T₀, T₁, and T₂.     

Integral equations and inequalities of Volterra type for functions defined in partially ordered spaces

The paper considers Volterra type integral operators acting in L₂(T), where T is a partially ordered topological space as well as equations and inequalities related to them. For the linear operators of this type it is shown that they are quasinilpotent. Explicit estimates for the solutions of linear integral inequalities have been obtained. Nonlinear equations and inequalities have also been considered.     

New separation axioms in generalized topological spaces

We introduce and study new separation axioms in generalized topological spaces, namely, m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}}, m-T_\frac38\mu \mbox{-}T_{\frac{3}{8}} and m-T_\frac12\mu\mbox{-}T_{\frac{1}{2}}. m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces are strictly placed between μ-T ₀ spaces and m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}}, m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces are strictly placed between m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces and m-T_\frac12\mu \mbox{-}T_{\frac{1}{2}} spaces, and m-T_\frac12\mu\mbox{-}T_{\frac{1}{2}} spaces are strictly placed between m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces and μ-T ₁ spaces.     

H-spectral spaces

Let X be a T₀-space, we say that X is H-spectral if its T₀-compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T₁-spaces the T₀-compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T₁-space X in order to get its Wallman compactification spectral.     

Filter spaces

The category FIL of filter spaces and cauchy maps is a topological universe. This paper establishes the foundation for a completion theory forT ₂ filter spaces.     

R1 fuzzy topological spaces

A fuzzy topological analog of the R₁ separation axiom of topology is introduced and its appropriateness is established.     

On the role of finite,hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces

We consider how properties of the bonding maps of the inverse spectrum determine properties of the inverse limit. Specifically, we study the limits of inverse spectra of finite T₀-spaces with bonding maps which are either chaining or normalizing. We will show that if the bonding maps are normalizing, then the inverse limit is a normal T₀-space, and therefore, its Hausdorff reflection is its subset of specialization minimal elements. If the maps are chaining, then the inverse limit is a completely normal spectral space; such spaces have been studied since they include the real spectra of commutative rings [C.N. Delzell, J.J. Madden, J. Algebra 169 (1994) 71], and the prime spectrum of a ring of functions, Spec(C(X)). The existence and importance of this class of non-Hausdorff, normal topological spaces was extremely surprising to us. Further, each of these results is reversible; if the inverse limit is normal, then each space in the spectrum is preceded by one whose bonding map to it is normalizing. By way of contrast, the inverse limit of finite T₀-spaces with separating bonding maps need not be a normal topological space (Example 3.8(a)) and furthermore, if the spaces of the inverse spectrum are normal, then the Hausdorff reflection of the limit must be zero-dimensional (Theorem 3.15).     

Certain properties of bitopological spaces

Some connections and interrelations between T_i-separation axioms (i=1,2,3) for bitopological spaces are considered. In particular, four different versions of the definition of the pairwise Hausdorff separation axion are discussed. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 135–140. Translated by O. A. Ivanov.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

C-scattered fuzzy topological spaces

In this paper, we define the concept of C-scattered fuzzy topological spaces and obtain some properties about them. In particular, we study the relation between C-scattered spaces and its fuzzy extension, it is proved that C-scattered fuzzy topological spaces are invariant by fuzzy perfect maps, and that, in the realm of paracompact fuzzy topological spaces, the C-scattered spaces verify that their product by other fuzzy spaces is also paracompact fuzzy.     

Topologically complete representations of inverse semigroups

A representation of an inverse semigroup by means of partial open homeomorphisms of a topological T₀-space is called topologically complete if the domains of these partial homeomorphisms form a base of the topology. It is shown how to construct topologically complete representations on the base of a ternary relation satisfying some elementary axioms. This result makes it possible to obtain a pseudo-elementary axiomatization for inverse semigroups that have faithful topologically complete representations in T₁, T₂ and T₃-spaces. A topology is introduced on any antigroup; this topology is a concomitant of the algebraic structure and every topologically complete representation is continuous with respect to this topology.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Fell topology on the space of functions with closed graph

LetX andY beT ₁ topological spaces andG(X, Y) the space of all functions with closed graph. Conditions under which the Fell topology and the weak Fell topology coincide onG(X,Y) are given. Relations between the convergence in the Fell topologyτF, Kuratowski and continuous convergence are studied too. Characterizations of a topological spaceX by separation axioms of (G(X, R), τF) and topological properties of (G(X, R), τF) are investigated.     

Antiproximinal sets in spaces of continuous functions

Closed convex bounded antiproximinal bodies are constructed in the infinite-dimensional spacesC(Q), C ₀(T), L(S, S, ), andB(S), whereQ is a topological space andT is a locally compact Hausdorff space. It is shown that there are no closed bounded antiproximinal sets in Banach spaces with the Radon-Nikodym property.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 643–657, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00196.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.     

A characterization of fuzzy compactifications

It is shown that if X is a fuzzy T₂-space, then X has a fuzzy T₂-compactification if and only if X is a weakly induced ultra completely regular space. Also, for an arbitrary fuzzy topological space, a characterization is given of the set of all ultra fuzzy Compactifications.     

S-Regular and S-Normal Spaces

In this paper s-regular and s-normal spaces are characterized using semi-T₀-identification spaces, topological sums of s-regular and s-normal spaces are examined, and the relationships between s-regular, s-normal, and other separation axioms are further examined.     

Compact and strictly singular operators on Orlicz spaces

If 0 < p < 1 andT: L_p(0,1) →E is a continuous linear operator into a topological vector space, there is an infinite-dimensional subspaceX ofL _p on whichT is an isomorphism; thus there are no compact operators onL _p . Results of this type are proved for general non-locally convex Orlicz spaces.