Closed equivalence relations on compact spaces and pairs of commutative C*-algebras: a Categorical Approach

As E. Manes [3] has shown, the se t based category of compact Hausdorff spaces and their continuous maps is isomorphic to the category of algebras for the ultrafilter monad on sets. Several proofs of this result have been given; they show clearly that the isomorphism between the categories of compact Hausdorff spaces and of ultrafilter monad algebras is closely tied in with the Axiom of Choice. Thus the question is legitimate: what happens if we replace by a topos? When trying to answer this question, we are immediately confronted with another question: how do we generalize ultrafilters? Should we use prime filters, or should we use uitrafilters as defined by H. Volger [8]? Should we replace non -emp ty by n o n initial or by inhabited? We have not been able to decide between the various possibilities; thus we shall use them on an equal footing. Our first task is to construct suitable objects and monads of prime filters and of ultrafilters; this is accomplished in Sections 2 and 3. In Section 2, we obtain submonads of the double powerset monad from propositional connectives. This allows us to construct a filter monad and a prime filter monad on a topos, as well as arithmetic lattices and Stone spaces based on a topos. In Section 3, we construct a sets of inhabited subsets monad from a contravariant adjunction. One part of this adjunction is a contravariant powerset functor, from the category of partial morphisms in a topos to the topos. One can easily show, following the method of Par~ [5], that this functor is monadic; we shall not do this here. Combining the results of these two sections, by taking intersections of submonads of the double powerset monad, we obtain all the prime filter and ultrafilter monads we may want. The algebraic side of the problem is thus in good shape; this cannot be said for the topological side which we discuss in Section 4. In this section, we define compact Hausdorff spaces in a topos, relative to a submonad 9 of the filter monad, and we obtain an induced algebra functor Jr, from compact Hausdorff spaces to T algebras. In the other direction, we construct an induced topology functor, from T algebras to topological spaces. Induced topologies of T-algebras need not be Hausdorff. If we restrict ourselves to algebras for which the induced topology is Hausdorff, then induced topologies define a right inverse left adjoint functor of the induced algebra functor J.. We use standard notations as much as possible. Additional notations and auxiliary results are collected in Section 1. In particular, we define characteristic functions of relations, and discuss some of their properties. This is one useful tool for the present paper; the Mitchell B~nabou Osius language is another. We use this language in the form presented by G. Osius in [4]. Most of the material presented in this paper is taken from the unpublished thesis of the f i rs t -named author [6].

We consider an alternate form of the equivalence between the category of compact Hausdorff spaces and continuous functions and a category formed from Gleason spaces and certain relations. This equivalence arises from the study of the projective cover of a compact Hausdorff space. This line leads us to consider the category of compact Hausdorff spaces with closed relations, and the corresponding subcategories with continuous and interior relations. Various equivalences of these categories are given extending known equivalences of the category of compact Hausdorff spaces and continuous functions with compact regular frames, de Vries algebras, and also with a category of Gleason spaces that we introduce. Study of categories of compact Hausdorff spaces with relations is of interest as a general setting to consider Gleason spaces, for connections to modal logic, as well as for the intrinsic interest in these categories.

Integral representation of belief measures on compact spaces

De Vries powers: A generalization of Boolean powers for compact Hausdorff spaces

Enriched Stone-type dualities

The aim of this research is to extend the new type of compact spaces called Q* compact spaces, study its properties and generate new results of the space. It investigate the Q*-compactness of topological spaces with separable, Q*-metrizable, Q*-Hausdorff, homeomorphic, connected and finite intersection properties. The closed interval [0, 1] is Q* compact. So, it is deduced that the closed interval [0, 1] is Q*-compact. For example, if (X, τ) = ℝ and A = (0, ∞) then A is not Q*-compact. A subset S of ℝ is Q*-compact. Also, if (X, τ) is a Q*-compact metrizable space. Then (X, τ) is separable. (Y, τ1) is Q*-compact and metrizable if f is a continuous mapping of a Q*-compact metric space (X, d) onto a Q*-Hausdorff space (Y, τ1). An infinite subset of a Q*-compact space must have a limit point. The continuous mapping of a Q*-compact space has a greatest element and a least element. Eleven theorems were considered and their results were presented accordingly.

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.

Let C denote the category of compact Hausdorff spaces and continuous maps and H: C-,.HC the homotopy functor to the homotopy category. Let S: C-..SC denote the functor of shape in the sense of Holsztynski for the projection functor H. Every continuous mapping f between spaces gives rise to a shape morphism S(f) in SC, but not every shape morphism is in the image of S. In this paper it is shown that if X is a continuum with x E X and A is a compact connected abelian topological group, then if F is a shape morphism from X to A, then there is a continuous map f:X-'.A such thatf(x)=O and S(f)=F. It is also shown that if f, g: X-A are continuous withf(x)=g(x)=O and S(f)=S(g), then fandg are homotopic. These results are then used to show that there are shape classes of continua containing no locally connected continua and no arcwise connected continua. Some other applications to shape theory are given also. Ihtroduction. Let C denote the category of compact Hausdorff spaces and continuous maps and H: C-iHC the homotopy functor to the homotopy category. Let S: C-).SC denote the functor of shape in the sense of Holsztyn'ski for the projection functor H [5]. Let X and Y be compact Hausdorff spaces. In [6] it is shown that if X and Y are associated with ANR-systems X and Y, respectively, then there is one to one correspondence between Morsc(X, Y) and the homotopy classes of maps of ANRsystems used in the approach of Mardesic and Segal [7]. Thus, our results in this paper will apply to either approach to shape. In the first part of the paper we show that if X is a continuum with x E X and A is a compact connected abelian topological group, then if Fe Morsc(X, A), then there is a continuous f: X-.A with S(f)=F and withf(x)=0. It is also shown that if X and A are as above andf, g:X-+A are continuous with f(x)=g(x)=O and with S(f)=S(g), then f and g are homotopic. These results are clearly related to the results in [6]. Received by the editors November 14, 1972. AMS (MOS) subject classfications (1970). Primary 55D99; Secondary 22B99.

We assume a basic knowledge of general topology and integration theory (as found, for example, in [91], [104], [130], and [253]). The purpose of this chapter is to present some special results which are not necessarily found in general references. In section 4 we prove an interpolation theorem and investigate when compact Hausdorff spaces can be mapped continously onto the closed unit interval [0, 1]. A brief development of dispersed spaces and their relationship to spaces of ordinal numbers is given in section 5. Section 6 is devoted to a study of the Cantor set and section 7 is concerned with extremally disconnected compact Hausdorff spaces and their role as projectives (in the category of compact Hausdorff spaces and continuous maps). In section eight we briefly develop the theory of regular Borel measures and prove representation theorems for C(T,ℝ)* (and C(T,ℂ)*).

It is well known that the category of compact Hausdorff spaces is dually equivalent to the category of commutative $$C^\star $$ -algebras. More generally, this duality can be seen as a part of a square of dualities and equivalences between compact Hausdorff spaces, $$C^\star $$ -algebras, compact regular frames and de Vries algebras. Three of these equivalences have been extended to equivalences between compact pospaces, stably compact frames and proximity frames, the fourth part of what will be a second square being lacking. We propose the category of bounded Archimedean $$\ell $$ -semi-algebras to complete the second square of equivalences and to extend the category of $$C^\star $$ -algebras.

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

We prove that the weak* topologization of the set of all idempotent measures (Maslov measures) on compact Hausdorff spaces defines a functor on the category of compact Hausdorff spaces, and this functor is normal in the sense of E. V. Shchepin; in particular, it has many properties in common with the probability measure functor and the hyperspace functor. Moreover, we establish that this functor defines a monad in the category , and prove that the idempotent measure monad contains the hyperspace monad as a submonad. For the space of idempotent measures there is an analogue of the Milyutin map (that is, of a continuous map of compact Hausdorff spaces which admits a regular averaging operator for spaces of continuous functions). Using the assertion of the existence of Milyutin maps for idempotent measures, we prove that the idempotent measure functor is open, that is, it preserves the class of open surjective maps. We also prove that, in contrast to the case of probability measure spaces, the correspondence assigning to any pair of idempotent measures the set of measures on their product which have the given marginals is not continuous.

We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.

In this paper we establish a general duality theorem for compact Hausdorff spaces being recognizable over certain pairs consisting of a commutative unital topological semiring and a closed proper prime ideal. Indeed, we utilize the concept of blueprints and their localization to prove that the category of compact Hausdorff spaces generated by such a pair can be dually embedded into the category of commutative unital semirings if the pair possesses sufficiently many covering polynomials.

Atoms in the lattice of covering operators in compact Hausdorff spaces