Topological Hull Research Articles

Cartesian Closedness, Extensionality and Bornological Universes

In this paper we describe the Cartesian closed topological hull, extensional topological hull and topological universe hull of both the category of closed and of open bornological universes.

Exponentiality for the construct of affine sets

The topological construct SSET of affine sets over the two-point set S contains many interesting topological subconstructs such as TOP, the construct of topological spaces, and CL, the construct of closure spaces. For this category and its subconstructs cartesian closedness is studied. We first give a classification of the subconstructs of SSET according to their behaviour with respect to exponenttiality. We formulate sufficient conditions implying that a subconstruct behaves similar to CL. On the other hand, we characterize a conglomerate of subconstructs with behaviour similar to TOP. Finally, we construct the cartesian closed topological hull of SSET.

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom ( X , Y ) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.

The quasitopos hull of the construct of closure spaces

<p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.</p>

The Cartesian Closed Topological Hull of the Category of Approach Uniform Spaces

The category AUnif of approach uniform spaces and uniform contractions properly combines uniform spaces and extended pseudo-metric spaces but (like Unif) lacks convenience, such as cartesian closedness. This paper therefore considers its cartesian closed topological hull, which is first described as a subcategory of SAULim, the category of semi-approach uniform limit spaces and uniform contractions. This hull is then also given a description inside the topological universe hull of AUnif and is shown to be a reasonable generalization of the corresponding hull of Unif. Furthermore, some referencing notes are provided with respect to similar results that can be obtained when starting from qAUnif (where symmetry assumptions are omitted).

Realizations of Topological Categories

There are functor-preordering-structured categories S(F,P), defined by the Prague School, in which every concrete category over a concretizable basecategory is realizable. Over nice basecategories there are realizations of all topological categories in some topological S(F,L). This gives rise for a new characterization of those concrete categories having a topological hull.

Categorical Properties of Probabilistic Convergence Spaces

The purpose of this paper is to discuss some categorical properties of probabilistic convergence spaces. Its main theses are: (1) the construct P-PrTop of probabilistic pretopological spaces is the extensional topological hull of the construct FTPcs of FT-diagonal probabilistic convergence spaces for every triangular norm T; (2) the construct P-PsTop of probabilistic pseudotopological spaces is the topological universe hull of FTPcs for every triangular norm T.

Cartesian closed extensions of the category of uniform spaces

This paper studies cartesian closed topological categories inside the category of uniform limit spaces which contain the category Unif of uniform spaces. This approach yields many of the known cartesian closed extensions of Unif as well as some new ones. Applications include some new descriptions of the cartesian closed topological hull of Unif, and a relatively simple characterization of the finest splitting uniformity.

Cartesian closed topological and monotopological hulls: A comparison

The theory of cartesian closed monotopological hull of a category can be developed in analogy to that of the cartesian closed topological hull. The main objective of this paper is to describe these hulls in terms of one another. On the local level, a comparison between the cartesian objects of a monotopological category and the cartesian objects of its MacNeille completion is carried out. The cartesian closed monotopological hulls of several well-known subcategories of Top are determined.

Topological improvements of categories of structured sets

Categories of structured sets often fail to have some desirable properties. They may even fail to have any interesting decently behaved full subcategories. But, under some natural assumptions (and disregarding purely set theoretic problems concerning ‘size’), it is always possible to embed them into nicely behaved topological categories, in particular each such category has: (1) a topological hull (= Mac Neille completion), (2) a certesian closed topological hull (= Antoine-cpmpletion), (3) a hereditary topological hull, (4) a concrete quasitopos hull (= Wyler completion). The purpose of this paper is to discuss these hulls and to provide several illuminating examples.

CARTESIAN CLOSED TOPOLOGICAL HULLS AS INJECTIVE HULLS

Abstract It is shown that (concretely) Cartesian closed topological hulls can be characterized as injective hulls in a rather natural setting. The characterization of locale hulls as injective hulls in the category of (meet-) semilattices by Bruns & Lakser and Born & Kimura constitutes a special case.

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.

CARTESIAN CLOSEDNESS AND THE MCNEILLE COMPLETION OF AN INITIALLY STRUCTURED CATEGORY

Abstract An adaptation of a theorem by Herrlich [5] shows that every initially structured category A can be fully embedded in a topological category AC, which is, in fact, a MacNeille completion of A. It is then shown that A is Cartesian closed if and only if AC is. Also developed is the notion of a Cartesian closed initially structured (CCIS) hull of a category. The theory of the CCIS hull is analogous to that of the Cartesian closed topological (CCT) hull. It is proved that a category has a CCT hull-if and only if it has a CCIS hull; and this allows the list of conditions equivalent to the existence of a CCT hull to be supplemented. Examples are given, drawn mainly from the various categories of binary relations.

Cartesian Closed Topological Hulls

It is shown in this paper that if a concrete category $\mathfrak {A}$ admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then $\mathfrak {A}$ admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.

CARTESIAN CLOSED COREFLECTIVE HULLS

Conditions are established under which a given class of objects in a topological category will have a Cartesian closed coreflective hull. The main theorem is used to discover new Cartesian closed topological categories and to unify a diversity of known special results. It also provides a mild criterion for the existence of Cartesian closed topological hulls.

Cartesian closed topological hulls

It is shown in this paper that if a concrete category A \mathfrak {A} admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then A \mathfrak {A} admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.

Topological Rings of Quotients and Rings Without Open Left Ideals

Simple locally compact rings without open left ideals were considered in [13] and general locally compact rings without open left ideals were studied extensively in [5] and [6]. We remove the hypothesis of local compactness and consider topological rings A without open left ideals but containing an open subring R. In section 4 we show that under these conditions A is completely determined by R. More precisely A can be identified with the topological ring of quotients C(R) introduced in [8]. As an R-module RA is topologically isomorphic to I*(RR), the topological injective hull of RR. The last statement was proved in [6] for A locally compact and R compact. Section 5 gives a characterization of those linearly topologized rings R that can be openly embedded into a ring A without open left ideals. In particular we shall show that the open left ideals form an idempotent ideal filter with quotient ring A. In section 6 we consider the class ℋ of all topological rings that can be openly embedded into a topologic...