By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing (...) with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

I present solutions to several questions of Paul Bankston [2] by means of another version of the ultracoproduct construction, and explain the relation of ultracoproduct of compact Hausdorff spaces to other constructions combining topology, algebra and logic.

We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products (...) of non-empty sets with at most n points are non-empty. (shrink)

We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

The concept of neutrosophic locally compact and neutrosophic compact open topology are introduced and some interesting propositions are discussed.

It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with (...) the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$. This generalizes both Gelfand duality and Jónsson-Tarski duality. (shrink)

We show that the statement “separable, countably compact, regular spaces are Baire” is deducible from a strictly weaker form than AC, namely, CAC . We also find some characterizations of the axiom of dependent choices.

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces isK4and the modal logic of weakly scattered Stone spaces isK4G. As a corollary, we obtain thatK4is also the modal logic of compact Hausdorff spaces andK4Gis the modal logic of weakly scattered compact Hausdorff spaces.

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is Π20 source in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework of Type-2 Theory of Effectivity as precisely the countably (...) based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an ω-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every ω-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff–Kuratowski theorem generalizes to all quasi-Polish spaces. (shrink)

For a signature L with at least one constant symbol, an L‐structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L‐structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier‐free L‐sentences, and therefore one can define a Stone topology on. This topology on generalizes the topology of the space of n‐marked groups. We introduce a natural ultrametric on, and show that the Stone topology on (...) coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n‐ary symbols for any ). As one of the applications of compactness of the Stone topology on, we prove compactness of certain classes of metric spaces in the Gromov‐Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact. (shrink)

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. (...) Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations. (shrink)

In set theory without the axiom of choice math formula, three-space type results for the Hahn-Banach property are provided. We deduce that for every Hausdorff compact scattered space K, the Banach space C of real continuous functions on K satisfies the continuous Hahn-Banach property in math formula. We also prove in math formula Rudin's theorem: “Radon measures on Hausdorff compact scattered spaces are discrete”.

On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary relation (...) (‘qualitative probability’) that is tied to _measure_. It expresses that one region is _smaller than or equal in size_ to another. Algebraic models of our theory are _separation_ _σ_-_algebras with qualitative probability_: \((\mathbb {B}, \ll, \preceq )\), where \(\mathbb {B}\) is a Boolean _σ_-algebra, ≪ is a separation relation on \(\mathbb {B}\), and ≼ is a qualitative probability on \(\mathbb {B}\). We show that from algebraic models of this kind we can, in an interesting class of cases, recover a compact Hausdorff topology _X_, together with a countably additive measure _μ_ on a _σ_-field of Borel subsets of that topology, and that \((\mathbb {B}, \ll, \preceq )\) is isomorphic to a ‘standard model’ arising out of the pair (_X_, _μ_). It follows from one of our main results that any closed ball in Euclidean space, \(\mathbb {R}^{n}\), together with Lebesgue measure arises in this way from a separation _σ_-algebra with qualitative probability. (shrink)

Let M be an MV-algebra and ΩM be the set of all σ -valuations from M into the MV-unit interval. This paper focuses on the characterization of MV-algebras using σ -valuations of MV-algebras and proves that a σ -complete MV-algebra is σ -regular, which means that a ≤ b if and only if v ≤ v for any v ∈ ΩM. Then one can introduce in a natural way a fuzzy topology δ on ΩM. The representation theorem forMV-algebras is established (...) by means of fuzzy topology. Some properties of fuzzy topology δ and its cut topology U are investigated. (shrink)

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.

We prove in set theory without the Axiom of Choice, that Rado’s selection lemma (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document}) implies the Hahn-Banach axiom. We also prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document} is equivalent to several consequences of the Tychonov theorem for compact Hausdorff spaces: in particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document} implies that every filter on a well orderable set is included in (...) a ultrafilter. In set theory with atoms, the “Multiple Choice” axiom implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document}. (shrink)

The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨Δ, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of (...) consequence associated with ⊢Δ coincides with ⊨Δ. (shrink)

We study invariant measures on compact Hausdorff spaces using finitary integral logic. For each compact Hausdorff space X and any family equation image of its continuous transformations, we find equivalent conditions for the existence of an equation image-invariant measure on X. We give two proofs of the existence of Haar measure on compact groups.

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev ≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev ≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of (...) quantifier rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥α for all ordinals α; of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ω. The results of this paper include: (i) every map of level ≥ω is co-elementary; (ii) the limit maps of an ω-indexed inverse system of maps of level ≥α are also of level ≥α; and (iii) if K is a co-elementary class, k ≥ k (K,K) = Lev ≥ k+1 (K,K), then Lev ≥ k (K,K) = Lev ≥ω (K,K). A space X ∈ K is co-existentially closed in K if Lev ≥ 0 (K, X) = Lev ≥ 1 (K, X). Adapting the technique of "adding roots," by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one. (shrink)

For a free filter F on $$\omega $$ ω, endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F }, where $$p_F\not \in \omega $$ p F ∉ ω, with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F. Spaces of the form $$N_F$$ N F constitute (...) the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z. Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. (shrink)

In this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras to (...) a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: [email protected]: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750& diss=y. (shrink)

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. (...) [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras. (shrink)

Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $, into the Esterle algebra of formal power series. (...) Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on X. (shrink)

In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein , we develop an approach to sheaf representations of MV-algebras which combines two techniques for the representation of MV-algebras devised by Filipoiu and Georgescu and by Dubuc and Poveda . Following Davey approach , we use a subdirect representation of MV-algebras that is based on local MV-algebras. This allowed us to obtain: a representation of any MV-algebras as MV-algebra of all global sections of a sheaf of local MV-algebras on (...) the spectruum of its prime ideals; a representation of MV-algebras, having the space of minimal prime ideals compact, as MV-algebra of all global sections of a Hausdorff sheaf of MV-chains on the space of minimal prime ideals, which is a Stone space; an adjunction between the category of all MV-algebras and the category of MV-algebraic spaces, where an MV-algebraic space is a pair , where X is a compact topological space and F is a sheaf of MV-algebras with stalks that are local. (shrink)

This paper develops a general theory of uniform probability for compact metric spaces. Special cases of uniform probability include Lebesgue measure, the volume element on a Riemannian manifold, Haar measure, and various fractal measures (all suitably normalized). This paper first appeared fall of 1990 in the Journal of Theoretical Probability, vol. 3, no. 4, pp. 611—626. The key words by which this article was indexed were: ε-capacity, weak convergence, uniform probability, Hausdorff dimension, and capacity dimension.

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and (...) continuum-sized or locally compact Hausdorff. We also show that every extension of $\mathsf {S5}$ is the modal logic of a subalgebra of $\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality. (shrink)

Suppose that fn is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that equation imageT fn < T f0. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that equation imagefn < f0. This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments (...) used are not themselves constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (shrink)

The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the (...) known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint, arXiv:1201.5221v1]. (shrink)

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of the Lascar-Pillay theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of the Peter–Weyl theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types (...) for bounded hyperimaginaries. (shrink)

In the present paper, we introduce the notions of quasi-Boolean algebras as the generalization of Boolean algebras. First we discuss the related properties of quasi-Boolean algebras. Second we define filters of quasi-Boolean algebras and investigate some properties of filters in quasi-Boolean algebras. We also show that there is a one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra. Then we investigate the prime filters and maximal filters of quasi-Boolean algebras, showing that the (...) prime filters of a quasi-Boolean algebra are precisely the maximal filters and the prime spectrum of a quasi-Boolean algebra is a compact Hausdorff topological space. Finally, we define and study the reticulation of a quasi-Boolean algebra. (shrink)

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of (...) regularity, and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these (...) applications are not necessarily direct or expected. (shrink)

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (...) any finitely accessible category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

A compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X \times X, … as basic relations. Let f: X→ Y be a proper surjective holomorphic map between complex spaces and set Xy:=f-1(y). We show that the set Ak,d:={y∈ Y: the number of d-dimensional components of Xy is compact complex spaces and f: X→ (...) Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree. (shrink)

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$ -computable low $_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high $_{n}$ -degrees are required to (...) present the homeomorphism type of $X_n$. Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. (shrink)

Let A be the category of all reduced compact complex spaces, viewed as a multi-sorted first order structure, in the standard way. Let U be a sub-category of A, which is closed under the taking of products and analytic subsets, and whose morphisms include the projections. Under the assumption that Th(U) is unidimensional, we show that Morley rank is equal to Noetherian dimension, in any elementary extension of U. As a result, we are able to show that Morley degree (...) is definable in Th(U), when Th(U) is unidimensional. (shrink)

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.

We say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausdorff locale, the second dual of the frame of opens gives the frame of opens of the colimit. In order to arrive at (...) this conclusion, we generalize the Hofmann–Mislove–Johnstone theorem and some results regarding the patch construction for stably locally compact locales. (shrink)

We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.