On the dimension of proximity spaces

Smirnov, Yu. M.

doi:10.1090/trans2/021/01

Cited by 9 publications

(17 citation statements)

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“…Of course min dim is monotonic, for arbitrary subspaces. Smirnov has shown [17] that dim is not monotonic for closed subspaces; and as it happens, the same example shows that Ind and a Ind are not monotonic for closed subspaces. Both dim and Ind are monotonic for C*-embedded [4] normal subspaces.…”

Section: For a Metric Space M If δD M -mentioning

confidence: 80%

“…I believe that the only serious investigation of the dimension theory of nonnormal spaces so far has been the concluding section of Smirnov's paper [17]. There the dimension function dim is defined, as the covering dimension with respect to the family of all finite normal coverings, and the decidedly imperfect analogy with δd is worked out.…”

Section: For a Metric Space M If δD M -mentioning

confidence: 99%

“…Both dim and Ind are monotonic for C*-embedded [4] normal subspaces. (For dim, [17]; for Ind, an easy exercise.) For a Ind this is an open problem.…”

Section: For a Metric Space M If δD M -mentioning

confidence: 99%

“…Covering dimension dim has been successfully generalized by Smirnov [17]; here we add to Smirnov's theory a generalization of Aleksandrov's theorem characterizing dim by separating ^-tuples of pairs (A i9 Bi) of disjoint closed sets by closed sets C; with empty intersection. The notion of min dim, mentioned in Part I [7], is formally defined: min dim X is the minimum of Id μX over all compatible uniformities μ. Equivalently, it is the minimum of dim Y over spaces Y containing X.…”

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confidence: 99%

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On finite-dimensional uniform spaces. II

Isbell¹

1962

Pacific J. Math.

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Introduction. The main subject of this paper is the [inductive* dimension δ Ind μX of uniform spaces μX. This is defined similarly to topological dimension Ind, but instead of separation one uses the notion of a set H, arbitrarily small uniform neighborhoods of which uniformly separate given sets A, B. For finite dimensional metric spaces M (i.e. the large dimension Δd M is finite) 8 Ind coincides with the covering dimensions Ad and δd. For general spaces μX we have 8 Ind μX ^ δd μX. For all known examples (including the examples for Δd Φ δd and, in compact spaces, Ind Φ dim) 8 Ind coincides with δd.The last section of the paper concerns the dimension theory of uniformisable spaces; it organizes alternative definitions and formulates problems, giving limited results on some of the problems. Covering dimension dim has been successfully generalized by Smirnov [17]; here we add to Smirnov's theory a generalization of Aleksandrov's theorem characterizing dim by separating ^-tuples of pairs (A i9 Bi) of disjoint closed sets by closed sets C; with empty intersection. The notion of min dim, mentioned in Part I [7], is formally defined: min dim X is the minimum of Id μX over all compatible uniformities μ. Equivalently, it is the minimum of dim Y over spaces Y containing X. The question when min dim X -dim X, i.e. when X cannot be embedded in a space of lower dimension, is stressed. The Lindelof property implies this, but the question is open for metrizable spaces and more generally for spaces admitting a complete uniformity.It is shown that every completely metrizable space can be homeomorphically embedded as a closed set in a countable product of finitedimensional polyhedra. Combined with results of [9] this means that every completely metrizable space is an inverse limit of polyhedra of the same or lower dimension. The question is still open whether a 1-dimensional completely metrizable space can be an inverse limit of discrete spaces.An announcement of the results on 8 Ind appeared in [8],

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Section: For a Metric Space M If δD M -mentioning

confidence: 80%

Section: For a Metric Space M If δD M -mentioning

confidence: 99%

“…Both dim and Ind are monotonic for C*-embedded [4] normal subspaces. (For dim, [17]; for Ind, an easy exercise.) For a Ind this is an open problem.…”

Section: For a Metric Space M If δD M -mentioning

confidence: 99%

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confidence: 99%

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On finite-dimensional uniform spaces. II

Isbell¹

1962

Pacific J. Math.

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“…• In the case that X is finite-dimensional, the pair (K, X) has the extension property if, and only if, 1 + dim K ≤ dim X, where dim K is the covering dimension of the topological space K (see [Smi,Theorem 9t]). …”

Section: Proofmentioning

confidence: 99%

Boundaries for Spaces of Holomorphic Functions on $\mathcal C(K)$

Acosta¹

2006

Publ. Res. Inst. Math. Sci.

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Coproducts of proximity spaces

Grzegrzolka

2019

Afr. Mat.

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In this paper, we introduce coproducts of proximity spaces. After exploring several of their basic properties, we show that given a collection of proximity spaces, the coproduct of their Smirnov compactifications proximally and densely embeds in the Smirnov compactification of the coproduct of the original proximity spaces. We also show that the dense proximity embedding is a proximity isomorphism if and only if the index set is finite. After constructing a number of examples of coproducts and their Smirnov compactifications, we explore several properties of the Smirnov compactification of the coproduct, including its relation to the Stone-Cech compactification, metrizability, connectedness of the boundary, and dimension. We finish with an example of a coproduct with the covering dimension 0 but the proximity dimension ∞.2010 Mathematics Subject Classification. 54E05, 54D35.

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On the dimension of proximity spaces

Cited by 9 publications

References 0 publications

On finite-dimensional uniform spaces. II

On finite-dimensional uniform spaces. II

Boundaries for Spaces of Holomorphic Functions on $\mathcal C(K)$

Coproducts of proximity spaces

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