A regular space which is not completely regular

Mysior, Adam

doi:10.1090/s0002-9939-1981-0601748-4

Cited by 9 publications

(17 citation statements)

References 3 publications

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“…Our main concern in this section is to apply the abstract theory developed in the previous sections to prove the existence of statistical solutions to the Navier-Stokes equations (54)- (55). For more specific discussions on the notion of statistical solutions for the Navier-Stokes equations, we refer the reader to [27,75,76,33] and also to the more recent paper [36].…”

Section: Applicationsmentioning

confidence: 99%

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Abstract framework for the theory of statistical solutions

Bronzi

Mondaini

Rosa

2016

Journal of Differential Equations

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Abstract. An abstract framework for the theory of statistical solutions is developed for general evolution equations, extending the theory initially developed for the three-dimensional incompressible Navier-Stokes equations. The motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique. Both concepts of statistical solution in trajectory space and in phase space are given, and the corresponding results of existence of statistical solution for the associated initial value problems are proved. The wide applicability of the theory is illustrated with the very incompressible Navier-Stokes equations, a reaction-diffusion equation, and a nonlinear wave equation, all displaying the property of global existence of weak solutions without a known result of global uniqueness.

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Section: Applicationsmentioning

confidence: 99%

“…Then there exists a projected statistical solution {ρ t } t∈I of the Navier-Stokes equations (54)- (55), associated with a U I -trajectory statistical solution, such that (i) The initial condition ρ t 0 = µ 0 holds; (ii) The function…”

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confidence: 99%

Abstract framework for the theory of statistical solutions

Bronzi

Mondaini

Rosa

2016

Journal of Differential Equations

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“…We modify the approach carried out in [2], which consists in a generalization of Mysior's example, compare [10]. Despite the fact that our arguments resemble those used in [2], we believe that this presentation is a bit simpler and enables us to construct some nonhomeomorphic examples, for example spaces X(λ, κ).…”

Section: On Mysior's Examplementioning

confidence: 99%

“…Our discussion focuses around a question: How can a completely regular space be extended by a point to only a regular space? Before A. Mysior's example, such a construction seemed quite complicated, compare [10] and [3]. R. Engelking included a description of Mysior's example in the Polish edition of his book [4, p. 55-56].…”

Section: Introductionmentioning

confidence: 99%

On regular but not completely regular spaces

Kalemba

Plewik

2019

Topology and its Applications

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“…This question was motivated by his previous result that every balanced quasi-metric space is a Tychonoff space [3, Corollary 3]. In this note we construct a quiet transitive quasi-uniformity that is complete in the sense of Doitchinov and is compatible with A. Mysior's noncompletely regular space [6].The definitions due to Doitchinov needed in this paper are given below; for further information on quasi-uniform spaces see [4]. …”

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confidence: 99%

A non-completely regular quiet quasi-uniformity

Fletcher¹,

Hejcman²,

Hunsaker³

1990

Proc. Amer. Math. Soc.

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Abstract.A complete quiet quasi-uniformity is constructed on A. Mysior's regular but not completely regular space. This answers a question raised by D. Doitchinov at the Prague conference on categorial topology in 1988.It is well known that the classical theory of completeness for uniform spaces cannot be extended satisfactorily to the class of all quasi-uniform spaces. In [ 1 ] D. Doitchinov introduced the class of quiet quasi-uniform spaces and gave a theory of completion for them. His theory of completion is as well behaved as the classical theory of completion for uniform spaces, and his class of quiet quasi-uniform spaces includes all uniform spaces and many of the most interesting quasi-uniform spaces. He developed topological properties of these spaces in [2], where he established that every quiet quasi-uniform space is a regular Hausdorff space and asked whether each quiet space is completely regular. This question was motivated by his previous result that every balanced quasi-metric space is a Tychonoff space [3, Corollary 3]. In this note we construct a quiet transitive quasi-uniformity that is complete in the sense of Doitchinov and is compatible with A. Mysior's noncompletely regular space [6].The definitions due to Doitchinov needed in this paper are given below; for further information on quasi-uniform spaces see [4].

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A regular space which is not completely regular

Abstract: Abstract. We give a simple example of a regular space which is not completely regular.

Cited by 9 publications

References 3 publications

Abstract framework for the theory of statistical solutions

Abstract framework for the theory of statistical solutions

On regular but not completely regular spaces

A non-completely regular quiet quasi-uniformity

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