Sobrification and bicompletion of totally bounded quasi-uniform spaces

Künzi, Hans-Peter A.; Brümmer, G. C. L.

doi:10.1017/s0305004100066597

Cited by 32 publications

(20 citation statements)

References 21 publications

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“…Hence a topological space is hereditarily compact if and only if its well-monotone covering quasi-uniformity coincides with its Pervin quasi-uniformity (compare [8], propositions 2-7 and 2-8). Therefore the last statement of Proposition 5 generalizes that part of theorem 3 in [22] considerably which says that if SP is the Pervin quasiuniformity of a hereditarily compact T 0 -space X, then (X, ^(^)) is the sobrification ofX It is well-known that the Pervin quasi-uniformity is the finest compatible totally bounded quasi-uniformity on a topological space (see [8], p. 28). Thus on an arbitrary topological space X each compatible totally bounded quasi-uniformity V is coarser than the well-monotone covering quasi-uniformity of X.…”

Section: H a N S -P E T E R A Kunzi And Nathalie Ferrariomentioning

confidence: 58%

“…It follows from Proposition 1 (c) that each of the filters of the form as defined in Proposition 1 (c) is a '^'*-Cauchy filter on X. Using this fact one can give another proof of the result (established in [22], corollary on p. 239) that whenever "V is a compatible totally bounded quasi- uniformity on a 7J,-space X, then the space (X, &~("K)) contains the sobrification of X as a subspace.…”

Section: H a N S -P E T E R A Kunzi And Nathalie Ferrariomentioning

confidence: 81%

“…We are now ready to formulate our characterization of the topological spaces that have a bicomplete fine quasi-uniformity. Let us note that the topological 7^-spaces admitting a bicomplete totally bounded quasi-uniformity are characterized in [22] as the strongly sober locally compact spaces. PROPOSITION On the other hand, if X has a non-trivial closed irreducible subset F such that for any entourage Ve'W there is an XEF satisfying F £ V~x(x), then by Proposition 1 (c) and 1 (a) there exists a "^*-Cauchy filter on X without ^"(T^"*)-limit point in X.…”

Section: (T/n T R 1 )^)Bmentioning

confidence: 99%

“…We also try to understand why, in general, a compatible quasiuniformity t o n a topological 7J,-space X cannot be extended to a compatible quasiuniformity on the sobrification of X, although such an extension is always possible if °U is totally bounded (see [22], lemma 6).…”

Section: The Sobrification As Bicompletionmentioning

confidence: 99%

“…Our next two results should be compared with theorem 3 of [22] where the corresponding question is studied for the Pervin quasi-uniformity. COROLLARY …”

Section: P(fx) = Q(fx) ^ Q(fy)+q(yx)mentioning

confidence: 99%

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Bicompleteness of the fine quasi-uniformity

Künzi¹,

Ferrario²

1991

Math. Proc. Camb. Phil. Soc.

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A characterization of the topological spaces that possess a bicomplete fine quasiuniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable 7^-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T t -spaces that do not admit a bicomplete quasi-uniformity.We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober. IntroductionIt is known that the Pervin quasi-uniformity of a topological space X is bicomplete if and only if X is a hereditarily compact quasi-sober space ([14], essentially corollary 3 -2). Naturally this result suggests the question under which conditions other wellknown canonical quasi-uniformities °U (defined on appropriate classes of topological spaces) are bicomplete. In this note we wish to consider this problem in the case that°U is the fine quasi-uniformity or the semi-continuous quasi-uniformity of an arbitrary topological space.In the first part of this note we prove that the fine quasi-uniformity of each quasipseudo-metrizable space and of each sober space is bicomplete. We remark that, on the other hand, while it seems to be unknown whether the fine quasi-uniformity of each quasi-pseudo-metrizable space is complete, it is shown in [17] that the fine quasi-uniformity of some well-known normal Hausdorif spaces is not complete (see also [19, 21] for related results). As a by-product of our investigations we get the result that the sobrification of a topological (7J,)-space X can be obtained by constructing the bicompletion of the well-monotone covering quasi-uniformity ofX In [13] the analogy between the separated Cauchy completion of a uniform space and the sobrification of a topological space is studied from the point of view of category theory. Our observation may help to explain the similarity between these two constructions from a different point of view.In the second part of this note we try to determine familiar conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete. In particular we show that the semi-continuous quasi-uniformity of a sober hereditarily countably metacompact space is bicomplete if and only if the space is hereditarily closed-complete.

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Section: H a N S -P E T E R A Kunzi And Nathalie Ferrariomentioning

confidence: 58%

Section: H a N S -P E T E R A Kunzi And Nathalie Ferrariomentioning

confidence: 81%

Section: (T/n T R 1 )^)Bmentioning

confidence: 99%

Section: The Sobrification As Bicompletionmentioning

confidence: 99%

“…Our next two results should be compared with theorem 3 of [22] where the corresponding question is studied for the Pervin quasi-uniformity. COROLLARY …”

Section: P(fx) = Q(fx) ^ Q(fy)+q(yx)mentioning

confidence: 99%

See 3 more Smart Citations

Bicompleteness of the fine quasi-uniformity

Künzi¹,

Ferrario²

1991

Math. Proc. Camb. Phil. Soc.

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On the MacNeille Completion of the Category of Partially Ordered Topological Spaces

Schauerte

1993

Mathematische Nachrichten

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Preordered topological spaces for which the order has a closed graph form a topological category. Within this category we identify the MacNeille completions (coinciding with the universal initial completions) of five monotopological subcategories, namely those of the To( TI, T2) preordered spaces and the (completely regular) partially ordered spaces. We also show that a functor due to L. NACHBIN from the quasi-uniform spaces to the preordered spaces preserves initial sources.

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(Strongly) Zero-Dimensional Partially Ordered Spaces

Nailana

2000

Papers in Honour of Bernhard Banaschewski

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Sobrification and bicompletion of totally bounded quasi-uniform spaces

Cited by 32 publications

References 21 publications

Bicompleteness of the fine quasi-uniformity

Bicompleteness of the fine quasi-uniformity

On the MacNeille Completion of the Category of Partially Ordered Topological Spaces

(Strongly) Zero-Dimensional Partially Ordered Spaces

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