## Locales and Formal Spaces (2002)

### BibTeX

@MISC{Hohti02localesand,

author = {Aarno Hohti},

title = {Locales and Formal Spaces},

year = {2002}

}

### OpenURL

### Abstract

ABSTRACT. We investigate the connection between the spatiality of locale products and the earlier studies of the author on the locally fine coreflection of the products of uniform spaces. After giving a historical introduction and indicating the connection between spatiality and the locally fine construction, we indicate how the earlier results directly solve the first of the two open problems announced in the thesis of T. Plewe. Finally, we establish a general isomorphism between the covering monoids of the localic product of topological (completely regular) spaces and the locally fine coreflection of the corresponding product of (fine) uniform spaces. Additionally, paper relates the recent studies on formal topology and uniform spaces by showing how the transitivity of covering relations corresponds to the locally fine construction.

### Citations

76 |
spaces
- Stone
- 1982
(Show Context)
Citation Context ...ry, and A ⊂ Xω is analytic, then |A, S|ω > ω implies |A, S|ω = 2ω . Remark 1: A basic example of a non-spatial product is given by Q × Q. This example was explicitly handled by Johnstone in his book (=-=[34]-=-, II 2.14), which appeared in 1982. Coincidentally, in the same year, the author had shown as a particular corollary to his results on supercompleteness that λ(FQ × FQ) ̸= F(Q ×Q). This followed from ... |

42 |
Atomless parts of spaces
- Isbell
(Show Context)
Citation Context ...nt field. Let us note that the link between the spatiality of the localic products of paracompact spaces and the corresponding product of fine (uniform) spaces was already established by J. Isbell in =-=[32]-=-. On the other hand, the connection between locales and so-called formal spaces as clearly indicated in the thesis of I. Sigstam([46]), led the author to discover the relation of the latter to (pre-)u... |

40 |
Uniform Spaces
- Isbell
- 1964
(Show Context)
Citation Context ...ete metric space is again complete. Isbell proved that for a uniform space µX, the uniform hyperspace H(µX) is complete if, and only if, the locally fine coreflection λµ is fine and X is paracompact (=-=[30]-=-). Thus, for such supercomplete spaces, the locally fine construction is sufficient for defining topology (open covers) combinatorially from uniform ones. 3. Locales. For a topological space X, the to... |

36 | The point of pointless topology - Johnstone - 1983 |

31 |
A solution of the decision problem for the Lewis systems S2 and S4 with an application to topology
- McKinsey
- 1941
(Show Context)
Citation Context ...ed set. This may be followed by 5) modal logics (see [13]), closing the circle with the equivalence between the modal system S4 and the closure operation in topology, well known since the 1930’s (see =-=[36]-=-).The Locally Fine Coreflection and Locales 10 Let {Ui} ∈ µ, and for each i, let {V i j } ∈ µ. Thus, (X, {Ui}) ∈ R, and for each i, we have (Ui, {Ui ∩V i j by the definition of R. By the condition un... |

27 |
Formal spaces, The
- FOURMAN, GRAYSON
- 1981
(Show Context)
Citation Context ... gives a collection of ‘pieces’ of a space related through a covering relation and studies recursive constructions on the basis of these pieces. After being introduced by Fourman and Grayson in 1982 (=-=[9]-=-) and made manifest by Sambin in 1987 ([45]), this approach to a point-free topology has been studied by several authors (see, e. g., Negri and Valentini [38] and Sigstam [46]). As will be seen in the... |

12 |
Grothendieck topology as geometric modality
- Goldblatt
- 1981
(Show Context)
Citation Context ...formal spaces, we may add 4) Grothendieck topologies, because the covering relation gives the conditions for a Grothendieck topology on a pre-ordered set. This may be followed by 5) modal logics (see =-=[13]-=-), closing the circle with the equivalence between the modal system S4 and the closure operation in topology, well known since the 1930’s (see [36]).The Locally Fine Coreflection and Locales 10 Let {... |

8 |
Intuitionistic formal spaces and their neighbourhood, Logic Colloquium ’88
- Sambin
- 1989
(Show Context)
Citation Context ... related through a covering relation and studies recursive constructions on the basis of these pieces. After being introduced by Fourman and Grayson in 1982 ([9]) and made manifest by Sambin in 1987 (=-=[45]-=-), this approach to a point-free topology has been studied by several authors (see, e. g., Negri and Valentini [38] and Sigstam [46]). As will be seen in the sequel, the transitivity requirement of th... |

6 | Sums in the category of frames
- Dowker, Strauss
- 1977
(Show Context)
Citation Context ...describe the locale product simply as the locally fine (or λ-) product. The product theorem given in this section grew out of the author’s attempt to understand the proof given by Dowker and Strauss (=-=[6]-=-) for their product theorem. The following definitions are well-known, see, e. g., [46]. Let Cov be a covering relation on a pre-ordered set P. For subsets U, V ⊂ P, define U ≤ V if for all u ∈ U, we ... |

6 |
Tychonoff’s theorem in the framework of formal topologies
- Negri, Valentini
- 1997
(Show Context)
Citation Context ...introduced by Fourman and Grayson in 1982 ([9]) and made manifest by Sambin in 1987 ([45]), this approach to a point-free topology has been studied by several authors (see, e. g., Negri and Valentini =-=[38]-=- and Sigstam [46]). As will be seen in the sequel, the transitivity requirement of the covering relation of a formal space is the counterpart of Kuratowski’s classical condition that C2 = C for a clos... |

5 |
Formal spaces and their effective presentations
- Sigstam
- 1995
(Show Context)
Citation Context ...t of fine (uniform) spaces was already established by J. Isbell in [32]. On the other hand, the connection between locales and so-called formal spaces as clearly indicated in the thesis of I. Sigstam(=-=[46]-=-), led the author to discover the relation of the latter to (pre-)uniform spaces defined by means of filters of coverings of a space. The transitivity of the covering relations of formal 1 During the ... |

4 | Minimal invariant spaces in formal topology
- Coquand
- 1997
(Show Context)
Citation Context ...s for topological spaces have been proved by several authors (see, e. g., Tychonoff’s Theorem in Negri and Valentini [38] and Coquand’s version of van der Waerden’s theorem on arithmetic progressions =-=[4]-=-.) Formal spaces were used in 1990 by Sigstam to give an effective theory of spaces in her thesis [46]. The approach is opposite (‘top-down’) to the traditional (‘bottom-up’) constructive approaches t... |

4 |
Gattungen von lokalen Strukturen, Jahresbericht der Deutschen
- Ehresmann
- 1957
(Show Context)
Citation Context ...sms are called local lattices (Benabou) or simply locales (Isbell). Thus, we have a contravariant functor T : Top → Loc. Very general ‘local structures’ satisfying (∗) were considered by Ehresmann in =-=[7]-=-, who also defined the notion of ‘paratopology’, studied in the papers [39], [3]. The category of frames is the opposite Loc op of the category of locales, i. e., they are complete right distributive ... |

4 |
A note on completely metrizable spaces
- Michael
- 1986
(Show Context)
Citation Context ... on spatial products, and also improved our theorem in [27] that Čech-scattered paracompacta have paracompact countable products by establishing that countable localic products of partition-complete (=-=[37]-=-, [49]) paracompact spaces are spatial. The connection to the locally fine coreflection was, however, never pointed out. Formal spaces form a counterpart of the original constructive approaches to top... |

3 |
Treillis locaux et paratopologies. Séminaire Ehresmann (Topologie et Géométrie Différentielle), 1re année (1957-8), exposé 2
- Bénabou
- 1958
(Show Context)
Citation Context ...paracompact, in the case of so-called supercomplete spaces. Their work [12] was published in 1959. However, the then recently introduced study of ‘local lattices’ and ‘paratopologies’ (see J. Benabou =-=[3]-=-, D. and S. Papert [39]) was taken up by Isbell resulting in an article on uniform locales [32], in which he characterized [12] as a paper “about objects in a category H now visible as the hypercomple... |

3 |
Localic products of spaces
- Plewe
- 1996
(Show Context)
Citation Context ...oducts of topological spaces and the relationship of the locally fine corecflection with the so-called formal spaces. Our work has been motivated by the appearance in 1996 of T. Plewe’s Ph. D. thesis =-=[41]-=- 2 on the spatiality of localic products of topological, especially separable metrizable spaces. Not only did the author discover an analogy between the previous results and those obtained in the thes... |

2 |
Sur les treillis des ouverts et les paratopologies, Séminaire C. Ehresmann de Topologie et de Géometrie Différentielle, 1957/58, Fac. des Sciences de
- Papert, Papert
- 1959
(Show Context)
Citation Context ...se of so-called supercomplete spaces. Their work [12] was published in 1959. However, the then recently introduced study of ‘local lattices’ and ‘paratopologies’ (see J. Benabou [3], D. and S. Papert =-=[39]-=-) was taken up by Isbell resulting in an article on uniform locales [32], in which he characterized [12] as a paper “about objects in a category H now visible as the hypercomplete uniform locales” (ib... |

2 |
On the notion of n-cardinality
- Przymusiński
- 1978
(Show Context)
Citation Context ...tting’ theorems for supercompleteness in products. The method was based on the notion of n-cardinality, due to T. Przymusinski and van Douwen, who gave similar applications to topological spaces (cf. =-=[43]-=-). Theorem 4. ([22]): For each n ∈ {1, 2, 3, . . .} there is a subset X ⊂ R such that F(X) k is supercomplete for k = 1, . . .,n but F(X) n+1 is not, in other words λ(F(X) n ) = F(X n ) but λ(F(X) n+1... |

1 |
A class of spaces whose Cartesian product with every hereditarily Lindelöf space is
- Alster
- 1981
(Show Context)
Citation Context ...e (see below) related to the operation λ. Wellknown topological cases known previously (the locally compact paracompact case in Arhangels’kii [2] and Frolík [11], C-scattered Lindelöf spaces (Alster, =-=[1]-=-), scattered paracompact spaces (Rudin, Watson [44]) and paracompact C-scattered spaces (Friedler, Martin, Williams [10])) all follow from the uniform case by suitably choosing the uniformity consider... |

1 |
On topological spaces complete in the sense of Čech, Vestnik
- Arhangel’skii
- 1961
(Show Context)
Citation Context ...pact spaces, made possible by the Noetherian tree technique (see below) related to the operation λ. Wellknown topological cases known previously (the locally compact paracompact case in Arhangels’kii =-=[2]-=- and Frolík [11], C-scattered Lindelöf spaces (Alster, [1]), scattered paracompact spaces (Rudin, Watson [44]) and paracompact C-scattered spaces (Friedler, Martin, Williams [10])) all follow from the... |

1 | Determination of paracompactness by uniformities - Corson - 1958 |

1 | Lindgren: C-complete quasi-uniform spaces - Fletcher, F - 1978 |

1 |
Paracompact C-scattered spaces
- Friedler, Martin, et al.
- 1987
(Show Context)
Citation Context ... case in Arhangels’kii [2] and Frolík [11], C-scattered Lindelöf spaces (Alster, [1]), scattered paracompact spaces (Rudin, Watson [44]) and paracompact C-scattered spaces (Friedler, Martin, Williams =-=[10]-=-)) all follow from the uniform case by suitably choosing the uniformity considered. The result of [26] was extended to Č-scattered paracompact spaces by the author and Yun Ziqiu in [27], first announc... |

1 |
On the topological product of paracompact spaces
- Frolík
- 1960
(Show Context)
Citation Context ...de possible by the Noetherian tree technique (see below) related to the operation λ. Wellknown topological cases known previously (the locally compact paracompact case in Arhangels’kii [2] and Frolík =-=[11]-=-, C-scattered Lindelöf spaces (Alster, [1]), scattered paracompact spaces (Rudin, Watson [44]) and paracompact C-scattered spaces (Friedler, Martin, Williams [10])) all follow from the uniform case by... |

1 |
Isbell: Some operators on uniform spaces
- Ginsburg, R
- 1959
(Show Context)
Citation Context ... obtained topology – for metric spaces – only in the case of complete spaces, or more generally, when the spaces considered were paracompact, in the case of so-called supercomplete spaces. Their work =-=[12]-=- was published in 1959. However, the then recently introduced study of ‘local lattices’ and ‘paratopologies’ (see J. Benabou [3], D. and S. Papert [39]) was taken up by Isbell resulting in an article ... |

1 | Some nearly fine uniform spaces - Hager - 1974 |

1 |
Die Schlichten stetigen Bilden des Nullraums, Fund
- Hausdorff
- 1937
(Show Context)
Citation Context ... a closed copy of of the irrationals J, because J is nowhere locally compact. On the other hand, if X is not C-scattered, then X contains a closed subspace F which is nowhere locally compact. Then by =-=[15]-=-, p. 157 (or ([41], 2.1), if X is completely metrizable, F contains a closed copy of J.) We note here that Isbell ([33], Th. 4) proved that in the class of completely regular spaces, X × loc Y is spat... |

1 |
On uniform paracompactness, Ann
- Hohti
(Show Context)
Citation Context ...bject Classification Numbers: Primary 54E15, Secondary 54B10. 1. Introduction. This paper 1 is based on the work carried out on the products of uniform spaces since 1981 by the author and others (see =-=[16]-=- – [24], [25], [26], [27]) using the technique of the locally fine corecflection. We study the connection of these results with the spatiality of localic products of topological spaces and the relatio... |

1 |
On supercomplete uniform spaces (II), Reports of the
- Hohti
- 1982
(Show Context)
Citation Context ... × F(Y ) is supercomplete for every paracompact space Y if, and only if, X is paracompact and C-scattered. This characterization was partially obtained in [21], and completed in [24]. The paper [21] (=-=[17]-=-) contained the result that F(X) × F(Y ) is supercomplete for every paracompact Y whenever X is C-scattered and paracompact. Furthermore, a partial converse obtained stated that if X is a paracompact ... |

1 |
On supercomplete uniform spaces
- Hohti
- 1987
(Show Context)
Citation Context ... every non-empty closed subspace of X contains a point with a closed neighbourhood which belongs to K. In the sequel, C (resp. Č) denotes the class of compact (resp. Čechcomplete) spaces. Theorem 1. (=-=[21]-=-, [24]): A binary product F(X) × F(Y ) is supercomplete for every paracompact space Y if, and only if, X is paracompact and C-scattered. This characterization was partially obtained in [21], and compl... |

1 |
Uniform hyperspaces, Colloquia Mathematica Societatis János Bolyai 41, Topology and Applications
- Hohti
- 1983
(Show Context)
Citation Context ...theoretical characterization. Noetherian trees can be used to give a direct motivation to a game-theoretical characterization of supercompleteness introduced – but not studied – by the author in 1983 =-=[19]-=-. There are two players I and II. For each game we choose an open cover V of the given uniform space µX. Player I begins by choosing a uniform cover U0 ∈ µ. If possible, Player II responds by selectin... |

1 |
On Ginsburg-Isbell derivatives and ranks of metric spaces
- Hohti
- 1984
(Show Context)
Citation Context ...= ΠPt(Li). 3 Ginsburg and Isbell ([12]) define µ (α+1) as µ (α) /µ (α) which, however, is less suitable for certain inductive purposes. The above ’slowed down’ version was introduced by the author in =-=[20]-=-. For products of the form µ × ν, we may still slow down the derivation by proceeding one coordinate at a time: Instead of using covers of the form U × V, where U ∈ µ and V ∈ ν, one considers covers U... |

1 |
On supercomplete uniform spaces III
- Hohti
(Show Context)
Citation Context ... supercompleteness in products. The method was based on the notion of n-cardinality, due to T. Przymusinski and van Douwen, who gave similar applications to topological spaces (cf. [43]). Theorem 4. (=-=[22]-=-): For each n ∈ {1, 2, 3, . . .} there is a subset X ⊂ R such that F(X) k is supercomplete for k = 1, . . .,n but F(X) n+1 is not, in other words λ(F(X) n ) = F(X n ) but λ(F(X) n+1 ) ̸= F(X n+1 ). In... |

1 |
On Relative ω–cardinality and Locally
- Hohti
- 1989
(Show Context)
Citation Context ...vious results and those obtained in the thesis in question, but the solution of the first open problem left open (re-solved consistently with ZFC) in [41] was seen to follow from the author’s article =-=[23]-=-. Therefore, we decided to write this paper in order to indicate the existence and usefulness of such previous research in an equivalent field. Let us note that the link between the spatiality of the ... |

1 |
On supercomplete spaces V: Tamano’s product problem
- Hohti
- 1990
(Show Context)
Citation Context ...lassification Numbers: Primary 54E15, Secondary 54B10. 1. Introduction. This paper 1 is based on the work carried out on the products of uniform spaces since 1981 by the author and others (see [16] – =-=[24]-=-, [25], [26], [27]) using the technique of the locally fine corecflection. We study the connection of these results with the spatiality of localic products of topological spaces and the relationship o... |

1 |
On complexity of metric spaces
- Hohti, Pelant
- 1985
(Show Context)
Citation Context ...ication Numbers: Primary 54E15, Secondary 54B10. 1. Introduction. This paper 1 is based on the work carried out on the products of uniform spaces since 1981 by the author and others (see [16] – [24], =-=[25]-=-, [26], [27]) using the technique of the locally fine corecflection. We study the connection of these results with the spatiality of localic products of topological spaces and the relationship of the ... |

1 |
On supercomplete uniform spaces IV: countable products, Fund
- Hohti, Pelant
- 1990
(Show Context)
Citation Context ...n Numbers: Primary 54E15, Secondary 54B10. 1. Introduction. This paper 1 is based on the work carried out on the products of uniform spaces since 1981 by the author and others (see [16] – [24], [25], =-=[26]-=-, [27]) using the technique of the locally fine corecflection. We study the connection of these results with the spatiality of localic products of topological spaces and the relationship of the locall... |

1 |
Yun Z: Countable products of Čech-scattered supercomplete spaces
- Hohti
(Show Context)
Citation Context ...ers: Primary 54E15, Secondary 54B10. 1. Introduction. This paper 1 is based on the work carried out on the products of uniform spaces since 1981 by the author and others (see [16] – [24], [25], [26], =-=[27]-=-) using the technique of the locally fine corecflection. We study the connection of these results with the spatiality of localic products of topological spaces and the relationship of the locally fine... |

1 |
The Locally Fine Coreflection and
- Hohti, Huˇsek, et al.
(Show Context)
Citation Context ...ΠXi) ∗ is analogous to the condition – studied by the author – that λ(ΠF(Xi)) contain all normal covers of ΠXi. (In [40] this was shown to be true whenever the Xi are completely metrizable spaces; in =-=[28]-=-, this result has been extended to paracompact spaces which are countable unions of closed, partition-complete subspaces.) Theorem 8.6: The localic product of a family (Xi) of regular topological spac... |

1 | Extensions and restrictions in products of metric spaces - Huˇsek, Pelant - 1987 |

1 | Product spaces in locales - Isbell - 1981 |

1 |
Locally fine uniformities and normal covers, Czechosl
- Pelant
- 1987
(Show Context)
Citation Context ... tree, the immediate successors of an element form its uniform cover, and the collection End(T) of all maximal elements of the tree T cover the space. This technique was first used in Pelant’s proof (=-=[40]-=-) 4 of Isbell’s conjecture that every locally fine space is ‘subfine’ (a subspace of a fine space). Pelant showed that the ‘λ equation’ considered above, λ(ΠF(Mi)) = F(ΠMi) holds for any collection of... |

1 | T:Countable products of absolute Cδ-spaces - Plewe - 1996 |

1 |
Countable products of scattered paracompact spaces
- Rudin, Watson
- 1983
(Show Context)
Citation Context ...n topological cases known previously (the locally compact paracompact case in Arhangels’kii [2] and Frolík [11], C-scattered Lindelöf spaces (Alster, [1]), scattered paracompact spaces (Rudin, Watson =-=[44]-=-) and paracompact C-scattered spaces (Friedler, Martin, Williams [10])) all follow from the uniform case by suitably choosing the uniformity considered. The result of [26] was extended to Č-scattered ... |

1 | H: Kernel constructions and Borel sets - Stone - 1963 |

1 |
C-scattered and paracompact spaces, Fund
- Telgársky
- 1971
(Show Context)
Citation Context ...eed, for metrizable spaces, the properties of 1) being C-scattered and 2) being completely metrizable and not containing a closed copy of the irrationals are equivalent. (If X is C-scattered, then by =-=[48]-=-, Theorem 1.7, it is an absolute Gδ space, and hence completely metrizable. The space X cannot contain a closed copy of of the irrationals J, because J is nowhere locally compact. On the other hand, i... |

1 |
Complete exhaustive sieves and games
- Telgársky, Wicke
- 1988
(Show Context)
Citation Context ...atial products, and also improved our theorem in [27] that Čech-scattered paracompacta have paracompact countable products by establishing that countable localic products of partition-complete ([37], =-=[49]-=-) paracompact spaces are spatial. The connection to the locally fine coreflection was, however, never pointed out. Formal spaces form a counterpart of the original constructive approaches to topology ... |