Countable products of absolute Cδ spaces

Plewe, Till

doi:10.1016/s0166-8641(96)00042-9

Cited by 6 publications

(5 citation statements)

References 19 publications

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“…(2) Do spatial multipliers for strongly Baire spaces have spatial countable localic powers? For completely regular spaces the answer to the first question is 'yes' [27]. But the only metrizable spaces which are contained in the class of absolute C 6 -spaces considered there are the completely metrizable spaces.…”

Section: Countable Products Of Non -Complete Spacesmentioning

confidence: 99%

“…Besides being intrinsically of interest, positive results on the preservation of products will have applications towards topology, since, as a rule, products are much better behaved in the category of locales than in the category of topological spaces. (The proof of the main theorem in [27] is an application of this type.) Compare, for example, the existing results on preservation of paracompactness under products in the category of topological spaces with the single theorem that products of paracompact regular locales are paracompact regular [15].…”

Section: Introductionmentioning

confidence: 99%

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Localic Products of Spaces

Plewe

1996

Proceedings of the London Mathematical Society

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The subject is spatiality of localic products of topological spaces, in particular metrizable spaces, or equivalently, preservation of products under the embedding of the category of sober spaces into the category of locales. A key theorem characterizes spatiality of (localic) products in terms of winning strategies of a strictly determined topological game. Further results concern metrizable spaces. The product of two metrizable spaces X1 and X2 is non‐spatial if and only if they have closed subspaces F1 and F2 respectively, which can be embedded into the Cantor set as dense subspaces with disjoint images. The locale X×loc Y is spatial for all Y if and only if X is complete and has no closed subspace homeomorphic to the irrationals. These spaces are called spatial multipliers. Using W. Hurewicz's characterization of strongly Baire spaces (every non‐empty closed subspace is of second category in itself) as spaces no closed subspace of which is homeomorphic to the rationals, we find that the characterization of spatial multipliers implies that a space which is not strongly Baire has spatial product only with the spatial multipliers. Accordingly for metrizable spaces the interesting questions lie within the strongly Baire spaces. The product of a complete space and a strongly Baire space is spatial. Spatiality of the localic product of two strongly Baire metrizable spaces implies that the topological product is strongly Baire. There is a Galois connection induced by the relation ‘X ∼ Y if and only if X×loc Y is spatial’. The poset of Galois‐closed classes contains subsets order isomorphic to the powerset of C−(C = 2N0). Finally, the question of absoluteness of Borel locales is considered, and it is shown that a metrizable locale which is Borel in every completion is spatial.

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Section: Countable Products Of Non -Complete Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Localic Products of Spaces

Plewe

1996

Proceedings of the London Mathematical Society

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“…But Πµ i has a basis consisting of products of finitely many locally finite open covers, and therefore so does λΠµ i , because the pre-uniformity This corollary can be stated also for other locally fine covering concepts, e. g., ultraparacompactness (refinement by clopen covers) etc. It was essentially proved by Plewe in [33]. He showed that the localic product of countably many partition-complete regular spaces is spatial.…”

Section: Countable Products Of σ-Partition-complete Supercomplete Spamentioning

confidence: 99%

“…For paracompact products, the above condition on the locally fine coreflection is equivalent to the spatiality of the corresponding product of locales, noted by Isbell in [24]. Spatiality was extensively studied by Plewe in [32], who used the productivity of paracompactness in locales and spatiality to extend ( [33]) the above topological results to countable products of C δ -absolute spaces. These spaces are equivalent to those with a complete exhaustive sieve studied by Michael in [28] and to the partition-complete spaces of Telgársky and Wicke in [42].…”

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The locally fine coreflection and normal covers in the products of partition-complete spaces

Hohti

Hušek

Pelant³

2002

Topology and its Applications

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We prove that the countable product of supercomplete spaces having a countable closed cover consisting of partition-complete subspaces is supercomplete with respect to its metric-fine coreflection. Thus, countable products of σ-partition-complete paracompact spaces are again paracompact. On the other hand, we show (Theorem 7.5) that in arbitrary products of partition-complete paracompact spaces, all normal covers can be obtained via the locally fine coreflection of the product of fine uniformities. These results extend those given in [1], [6], [7], [19], [30], [20], [33].

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“…seems to be rather difficult. Results concerning preservation of products (and some applications) can be found in [2,7,8]. Here we consider preservation of directed inverse limits; in particular Isbell's question whether a directed inverse limit of (quasi-)compact spatial locales is again spatial [1, p. 24].…”

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Directed inverse limits of spatial locales

Plewe

2002

Proc. Amer. Math. Soc.

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Abstract. In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.The question "Which limits are preserved by the embedding of sober topological spaces into the category of locales?", or equivalently "When is a limit of spatial locales spatial?" seems to be rather difficult. Results concerning preservation of products (and some applications) can be found in [2,7,8]. Here we consider preservation of directed inverse limits; in particular Isbell's question whether a directed inverse limit of (quasi-)compact spatial locales is again spatial [1, p. 24]. It is easy to see that if all locales are Hausdorff (i.e. if they have closed diagonals), then this question has a positive answer because (i) compact Hausdorff locales are spatial and closed under products, (ii) limits can be constructed as equalizers of a pair of maps between products of the original objects, and (iii) equalizers of maps between Hausdorff locales are closed inclusions and compactness is inherited by closed sublocales. So only the case of non-Hausdorff locales is problematic.This note consists of two parts. In the first part we give a construction which shows that directed inverse limits of compact locales need not be spatial even if all locales are furthermore locally compact and all maps open. However, in the second part we show that if all locales are locally compact and all maps proper (a weaker property suffices), then the inverse limit is again locally compact and hence spatial; here compactness is not needed.Constructions of the limit of a directed inverse system [6] or [10]. We don't need to go into any detail here since we will only be using the fact that if {B j } is a collection of bases for the locales X j , then the union of the inverse images under the maps f * ∞j is a base for the limit. This union is a subbase for any limit. But directedness implies that it is in fact a base.

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Countable products of absolute Cδ spaces

Cited by 6 publications

References 19 publications

Localic Products of Spaces

Localic Products of Spaces

The locally fine coreflection and normal covers in the products of partition-complete spaces

Directed inverse limits of spatial locales

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