PreT 2 Objects in Topological Categories

In previous papers, two notions of pre-Hausdorff (PreT ₂) objects in a topological category were introduced and compared. The main objective of this paper is to show that the full subcategory of PreT ₂ objects is a topological category and all of T ₀, T ₁, and T ₂ objects in this topological category are equivalent. Furthermore, the characterizations of pre-Hausdorff objects in the categories of filter convergence spaces, (constant) local filter convergence spaces, and (constant) stack convergence spaces are given and as a consequence, it is shown that these categories are homotopically trivial.     

CONCRETE CATEGORIES FOR WHICH FIBRE COMPLETIONS INDUCE TOPOLOGICAL COMPLETIONS

Abstract

Every topological category over an arbitrary base category X may be considered as a category of T-models with respect to some theory (i.e., functor) T from X into a category of complete lattices. Using this model-theoretic correspondence as our basic tool, we study initial and final completions of (co)fibration complete categories. For an arbitrary concrete category (A, U) over X, the process of order-theoretically completing each fibre does not usually yield an initial/final completion of (A, U). It is shown in this paper that for concrete categories which are assumed to be fibration and/or cofibration complete, initial and final completions can be constructed by completing the fibres. These completions are further shown to exhibit some interesting external properties.     

Quotient-reflective and bireflective subcategories of the category of preordered sets

In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T₀, T₁, and T₂ preordered sets and show that each of the full subcategories of each of T₀, T₁, T₂ preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord.     

ON CERTAIN INITIAL COMPLETIONS IN FUZZY TOPOLOGY

Abstract

Following a lead given by I.W. Alderton, it is shown that the MacNeille completion and the universal initial completion coincide for the categories of zero-dimensional fuzzy T₀-topological spaces, T₀-fuzzy closure spaces, 2T ₀-fuzzy bitopological spaces, and T ₁-fuzzy topological spaces and that these turn out to be respectively the categories of zero-dimensional fuzzy topological spaces, fuzzy closure spaces, fussy bitopological spaces, and fuzzy R ₀ topological spaces.     

On Q-sobriety

The study of fixed-basis variety-based topology was initiated by S.A. Solovyov (in 2008), which, among other things, generalizes fuzzy topology. We extend within this framework, an earlier result due to Srivastava et al. (in 1998), which showed that the category of sober fuzzy topological spaces is the epireflective hull of the fuzzy Sierpinski space in the category of T₀-fuzzy topological spaces.     

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.     

Normally Preordered Spaces and Utilities

In applications it is useful to know whether a topological preordered space is normally preordered. It is proved that every k _ω -space equipped with a closed preorder is a normally preordered space. Furthermore, it is proved that second countable regularly preordered spaces are perfectly normally preordered and admit a countable utility representation.     

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).     

On Homeomorphisms of Effective Topological Spaces

We study effective presentations and homeomorphisms of effective topological spaces. By constructing a functor from the category of computable models into the category of effective topological spaces, we show in particular that there exist homeomorphic effective topological spaces admitting no hyperarithmetical homeomorphism between them and there exist effective topological spaces whose autohomeomorphism group has the cardinality of the continuum but whose only hyperarithmetical autohomeomorphism is trivial. It is also shown that if the group of autohomeomorphisms of a hyperarithmetical topological space has cardinality less than 2 then this group is hyperarithmetical. We introduce the notion of strong computable homeomorphism and solve the problem of the number of effective presentations of T ₀-spaces with effective bases of clopen sets with respect to strong homeomorphisms.     

On epireflective and coreflective hulls of a particular L-topological space

In this paper, we study some aspects of the category L-ZTop of zero-dimensional L-topological spaces. After noting that it is a topological category, we identify a ‘Sierpinski object’ L_Z in it. We further show that two epireflective hulls of L_Z respectively turn out to be the categories of zero-dimensional T₀-L-topological spaces and of zero-dimensional sober L-topological spaces. We also determine the coreflective hull of L_Z in the category of L-topological spaces.     

Completion of merotopic spaces and extension of uniformly continuous maps

A general Riesz merotopic space (X, ν) determines a not necessarily topological closure operator c_ν on X. The space (X, ν) is said to be complete if every cluster on (X, ν) is contained in an adherence grill on (X, c_ν). We discuss a method of obtaining a large class of completions of a given Riesz merotopic space with induced T₁ closure space. As special cases we get the simple completion, which induces a simple closure space extension, and the strict completion, which induces a strict closure space extension. We show that the category of complete separated T₁ Riesz merotopic spaces is epireflective in the category of separated T₁ Riesz merotopic spaces, the reflection of an object being the simple completion. Similarly the category of complete clan-covered quasi-regular T₁ Riesz merotopic spaces is epireflective in the category of clan-covered quasi-regular T₁ Riesz merotopic spaces, the reflection of an object being the strict completion.     

A conglomerate of exponential supercategories of the category of finitely generated topological spaces

Summary For every ordinaln > 1 we define a categoryT _n of topological spaces in ech's sense which is isomorphic to a category ofn-ary monorelational systems. We show that every categoryT _n is an exponential supercategory of the categoryB of finitely generated topological spaces, which means that well-behaved function spacesG ^H can be defined inT _n wheneverG B.     

One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.     

Realization of cartesian closed topological hulls

The cartesian closed topological (CCT) hull of a concrete categoryK is the smallest full CCT extension ofK. A general method for describing the CCT hulls is presented and applied to render unified proofs for the basic examples ofK: topological spaces, uniform spaces, pretopological spaces, compact T₂ spaces, metrizable spaces, and completely regular spaces.     

DISCONNECTEDNESS

Abstract

There are three different ways to characterize To-spaces in the category of topological spaces. All three methods are canonical, i.e. they can be easily formulated in a general setting, where they, in general, do not coincide. In the following, the characterization of T₀-spaces by indiscrete spaces is generalized to an abstract category and investigated.     

A note on Cauchy spaces

In previous papers, various notions of (strongly) closed subobject, (strongly) open subobject, connected, compact and T _i , i=0,1,2 objects in a topological category were introduced and compared. The main objective of this paper is to characterize each of these classes of objects in the category of Cauchy spaces as well as to examine how these generalizations are related.