In this paper a systematic study of the category GTS of generalized topological spaces (in the se... more In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. Some completeness and cocompleteness results are achieved. Generalized topological spaces help to reconstruct the important elements of the theory of locally definable and weakly definable spaces in the wide context of weakly topological structures.
We prove three new versions of Stone Duality. The main version is the following: the category of ... more We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. Some theory of strongly locally spectral spaces is developed.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
We develop the theory of Heyting locally small spaces, including Stone-like dualities such as a n... more We develop the theory of Heyting locally small spaces, including Stone-like dualities such as a new version of Esakia duality and a system of concrete isomorphisms and equivalences. In such a way, we continue building tame topology, realising Grothendieck's ideas. We use up-spectral spaces and define the standard up-spectralification of a Kolmogorov locally small space. This research gives more understanding of locally definable spaces over structures with topologies.
In this paper a systematic study of the category GTS of generalized topological spaces (in the se... more In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. Some completeness and cocompleteness results are achieved. Generalized topological spaces help to reconstruct the important elements of the theory of locally definable and weakly definable spaces in the wide context of weakly topological structures.
In this paper a systematic study of the category GTS of generalized topological spaces (in the se... more In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. It is then applied to some topological constructions over first order topological structures. 2000 MS Classification: 54A05, 18F10, 03C65.
We develop the theory of Heyting small spaces and prove a new version of Esakia duality for such ... more We develop the theory of Heyting small spaces and prove a new version of Esakia duality for such spaces. To do this, we notice that spectral spaces may be seen as sober small spaces with all smops compact and introduce the method of the standard spectralification. This helps to understand open continuous definable mappings between definable spaces over o-minimal structures.
A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions... more A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.
Hu's metrization theorem for bornological universes is shown to hold in ZF and it is adapted to a... more Hu's metrization theorem for bornological universes is shown to hold in ZF and it is adapted to a quasi-metrization theorem for bornologies in bitopological spaces. The problem of uniform quasi-metrization of quasi-metric bornological universes is investigated. Several consequences for natural bornologies in generalized topological spaces in the sense of Delfs and Knebusch are deduced. Some statements concerning (uniform)-(quasi)-metrization of bornologies are shown to be relatively independent of ZF. A bitopological space is a triple (X, τ 1 , τ 2) where X is a set and τ 1 , τ 2 are topologies in X. A quasi-pseudometric in a set X is a function d : X × X → [0; +∞) such that, for all x, y, z ∈ X, d(x, y) ≤ d(x, z) + d(z, y) and d(x, x) = 0. A quasi-pseudometric d in X is called a quasi-metric if, for all x, y ∈ X, the condition d(x, y) = 0 implies x = y (cf. [Kel], [FL]). Let d be a quasi-pseudometric in X. The conjugate of d is the quasipseudometric d −1 defined by d −1 (x, y) = d(y, x) for x, y ∈ X. The d-ball with centre x ∈ X and radius r ∈ (0; +∞) is the set B d (x, r) = {y ∈ X : d(x, y) < r}. The collection τ (d) = {V ⊆ X : ∀ x∈V ∃ n∈ω B d (x, 1 2 n) ⊆ V } is the topology in X induced by d. The triple (X, τ (d), τ (d −1)) is the bitopological space associated with d. Definition 1.1. A bitopological space (X, τ 1 , τ 2) is (quasi)-metrizable if there exists a (quasi)-metric d in X such that τ 1 = τ (d) and τ 2 = τ (d −1) (cf. pp. 74-75 of [Kel]). One can find a considerable number of quasi-metrization theorems in [An] and in other sources (cf. [FL]). We recall that, according to [Al]-[Hu], a boundedness in a set X is a (non-void) ideal of subsets of X. A boundedness B in X is called a bornology in X if every singleton of X is a member of B (cf. 1.1.1 in [H-N]). Definition 1.2 (cf. Definition 4.1 of [Hu]). If B is a boundedness in X, then a collection A is called a base for B if A ⊆ B and every set of B is a subset of a member of A. A second-countable boundedness is a boundedness which has a countable base. Definition 1.3. Let (X, τ 1 , τ 2) be a bitopological space. A boundedness B in X will be called (τ 1 , τ 2)-proper if, for each A ∈ B, there exists B ∈ B such that cl τ 2 A ⊆ int τ 1 (B). If τ = τ 1 = τ 2 and the boundedness B is (τ, τ)-proper, we will say that B is τ-proper.
We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a b... more We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a branched covering is given. If the restriction π V : V → Y of the natural projection π : Y × Z → Y to a closed set V ⊂ Y × Z is a branched covering then, under certain assumptions, we can obtain generators of the fundamental group π 1 ((Y × Z) \ V).
Assuming ZF and its consistency, we study some topological and geometrical properties of the symm... more Assuming ZF and its consistency, we study some topological and geometrical properties of the symmetrized max-plus algebra in the absence of the axiom of choice in order to discuss the minimizing vector theorem for finite products of copies of the symmetrized max-plus algebra. Several relevant statements that follow from the axiom of countable choice restricted to sequences of subsets of the real line are shown. Among them, it is proved that if all simultaneously complete and connected subspaces of the plane are closed, then the real line is sequential. A brief discussion about semidenrites is included. Older known proofs in ZFC of several basic facts relevant to proximinal and Chebyshev sets in metric spaces are replaced by new proofs in ZF. It is proved that a nonempty subset C of the symmetrized max-plus algebra is Chebyshev in this algebra if and only if C is simultaneously closed and connected. An application of it to a version of the minimizing vector theorem for finite product...
We prove that the category of Kolgomorov small spaces and continuous mappings is equivalent to th... more We prove that the category of Kolgomorov small spaces and continuous mappings is equivalent to the category of spectral spaces with decent subsets and spectral mappings respecting the decent subsets and dually equivalent to the category of bounded distributive lattices with decent sets of prime filters and bounded lattice homomorphisms respecting those decent sets. This allows producing spectralifications of a Kolgomorov topological space using bounded sublattice bases of the topology.
We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for... more We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for locally small spaces with bounded continuous mappings.
In this paper, we prove new versions of Stone Duality. The main version is the following: the cat... more In this paper, we prove new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of (quasi-) compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms. Furthermore, it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. This helps to understand Kolmogorov locally small spaces and morphisms between them. We comment also on spectralifications of topological spaces.
Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on... more Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on X is defined by: ?* = {0}?{X\M:M is compact in (X,?)}. In this paper, properties of the space (X,?*) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.
In this paper a systematic study of the category GTS of generalized topological spaces (in the se... more In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. Some completeness and cocompleteness results are achieved. Generalized topological spaces help to reconstruct the important elements of the theory of locally definable and weakly definable spaces in the wide context of weakly topological structures.
We prove three new versions of Stone Duality. The main version is the following: the category of ... more We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. Some theory of strongly locally spectral spaces is developed.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
We develop the theory of Heyting locally small spaces, including Stone-like dualities such as a n... more We develop the theory of Heyting locally small spaces, including Stone-like dualities such as a new version of Esakia duality and a system of concrete isomorphisms and equivalences. In such a way, we continue building tame topology, realising Grothendieck's ideas. We use up-spectral spaces and define the standard up-spectralification of a Kolmogorov locally small space. This research gives more understanding of locally definable spaces over structures with topologies.
In this paper a systematic study of the category GTS of generalized topological spaces (in the se... more In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. Some completeness and cocompleteness results are achieved. Generalized topological spaces help to reconstruct the important elements of the theory of locally definable and weakly definable spaces in the wide context of weakly topological structures.
In this paper a systematic study of the category GTS of generalized topological spaces (in the se... more In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. It is then applied to some topological constructions over first order topological structures. 2000 MS Classification: 54A05, 18F10, 03C65.
We develop the theory of Heyting small spaces and prove a new version of Esakia duality for such ... more We develop the theory of Heyting small spaces and prove a new version of Esakia duality for such spaces. To do this, we notice that spectral spaces may be seen as sober small spaces with all smops compact and introduce the method of the standard spectralification. This helps to understand open continuous definable mappings between definable spaces over o-minimal structures.
A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions... more A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.
Hu's metrization theorem for bornological universes is shown to hold in ZF and it is adapted to a... more Hu's metrization theorem for bornological universes is shown to hold in ZF and it is adapted to a quasi-metrization theorem for bornologies in bitopological spaces. The problem of uniform quasi-metrization of quasi-metric bornological universes is investigated. Several consequences for natural bornologies in generalized topological spaces in the sense of Delfs and Knebusch are deduced. Some statements concerning (uniform)-(quasi)-metrization of bornologies are shown to be relatively independent of ZF. A bitopological space is a triple (X, τ 1 , τ 2) where X is a set and τ 1 , τ 2 are topologies in X. A quasi-pseudometric in a set X is a function d : X × X → [0; +∞) such that, for all x, y, z ∈ X, d(x, y) ≤ d(x, z) + d(z, y) and d(x, x) = 0. A quasi-pseudometric d in X is called a quasi-metric if, for all x, y ∈ X, the condition d(x, y) = 0 implies x = y (cf. [Kel], [FL]). Let d be a quasi-pseudometric in X. The conjugate of d is the quasipseudometric d −1 defined by d −1 (x, y) = d(y, x) for x, y ∈ X. The d-ball with centre x ∈ X and radius r ∈ (0; +∞) is the set B d (x, r) = {y ∈ X : d(x, y) < r}. The collection τ (d) = {V ⊆ X : ∀ x∈V ∃ n∈ω B d (x, 1 2 n) ⊆ V } is the topology in X induced by d. The triple (X, τ (d), τ (d −1)) is the bitopological space associated with d. Definition 1.1. A bitopological space (X, τ 1 , τ 2) is (quasi)-metrizable if there exists a (quasi)-metric d in X such that τ 1 = τ (d) and τ 2 = τ (d −1) (cf. pp. 74-75 of [Kel]). One can find a considerable number of quasi-metrization theorems in [An] and in other sources (cf. [FL]). We recall that, according to [Al]-[Hu], a boundedness in a set X is a (non-void) ideal of subsets of X. A boundedness B in X is called a bornology in X if every singleton of X is a member of B (cf. 1.1.1 in [H-N]). Definition 1.2 (cf. Definition 4.1 of [Hu]). If B is a boundedness in X, then a collection A is called a base for B if A ⊆ B and every set of B is a subset of a member of A. A second-countable boundedness is a boundedness which has a countable base. Definition 1.3. Let (X, τ 1 , τ 2) be a bitopological space. A boundedness B in X will be called (τ 1 , τ 2)-proper if, for each A ∈ B, there exists B ∈ B such that cl τ 2 A ⊆ int τ 1 (B). If τ = τ 1 = τ 2 and the boundedness B is (τ, τ)-proper, we will say that B is τ-proper.
We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a b... more We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a branched covering is given. If the restriction π V : V → Y of the natural projection π : Y × Z → Y to a closed set V ⊂ Y × Z is a branched covering then, under certain assumptions, we can obtain generators of the fundamental group π 1 ((Y × Z) \ V).
Assuming ZF and its consistency, we study some topological and geometrical properties of the symm... more Assuming ZF and its consistency, we study some topological and geometrical properties of the symmetrized max-plus algebra in the absence of the axiom of choice in order to discuss the minimizing vector theorem for finite products of copies of the symmetrized max-plus algebra. Several relevant statements that follow from the axiom of countable choice restricted to sequences of subsets of the real line are shown. Among them, it is proved that if all simultaneously complete and connected subspaces of the plane are closed, then the real line is sequential. A brief discussion about semidenrites is included. Older known proofs in ZFC of several basic facts relevant to proximinal and Chebyshev sets in metric spaces are replaced by new proofs in ZF. It is proved that a nonempty subset C of the symmetrized max-plus algebra is Chebyshev in this algebra if and only if C is simultaneously closed and connected. An application of it to a version of the minimizing vector theorem for finite product...
We prove that the category of Kolgomorov small spaces and continuous mappings is equivalent to th... more We prove that the category of Kolgomorov small spaces and continuous mappings is equivalent to the category of spectral spaces with decent subsets and spectral mappings respecting the decent subsets and dually equivalent to the category of bounded distributive lattices with decent sets of prime filters and bounded lattice homomorphisms respecting those decent sets. This allows producing spectralifications of a Kolgomorov topological space using bounded sublattice bases of the topology.
We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for... more We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for locally small spaces with bounded continuous mappings.
In this paper, we prove new versions of Stone Duality. The main version is the following: the cat... more In this paper, we prove new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of (quasi-) compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms. Furthermore, it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. This helps to understand Kolmogorov locally small spaces and morphisms between them. We comment also on spectralifications of topological spaces.
Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on... more Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on X is defined by: ?* = {0}?{X\M:M is compact in (X,?)}. In this paper, properties of the space (X,?*) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.
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