Note on hit-and-miss topologies

Zsilinszky, László

doi:10.1007/bf02904242

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…The following theorem is a generalization of classical results holding for hit and miss hypertopologies, [12,52]. In [52], L. Zsilinszki considers spaces that are weakly R 0 , i.e., every nonempty difference of open sets contains a non-empty closed subset of X. We will use an analogous property that holds for regular open and regular closed sets.…”

Section: Consider Now τ ⩕mentioning

confidence: 98%

Strong Proximities on Smooth Manifolds and Vorono\" i Diagrams

Peters¹,

Guadagni²

2015

Preprint

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This article introduces strongly near smooth manifolds. The main results are (i) second countability of the strongly hit and far-miss topology on a family B of subsets on the Lodato proximity space of regular open sets to which singletons are added, (ii) manifold strong proximity, (iii) strong proximity of charts in manifold atlases implies that the charts have nonempty intersection. The application of these results is given in terms of the nearness of atlases and charts of proximal manifolds and what are known as Voronoï manifolds.

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Section: Consider Now τ ⩕mentioning

confidence: 98%

Strong Proximities on Smooth Manifolds and Vorono\" i Diagrams

Peters¹,

Guadagni²

2015

Preprint

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“…The reader should note that the equivalence (2 ⇔ 3) in Theorem 5 can be viewed as a consequence of a general result of Zsilinszky [30,Theorem 2.3].…”

Section: Lemma 7 Let X Be a Topological Space In Which All Saturated mentioning

confidence: 99%

Fell topology on the hyperspace of a non-Hausdorff space

Hou

Vitolo

2008

Ricerche mat.

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Fell topology is very widely used today, even in metric spaces; but J. Fell introduced it in a non-Hausdorff context in the connection with the theory of C*-algebras. In spite of this, it has been studied only on the hyperspace of a Hausdorff space, except for the first results due to Fell himself. The present paper aims to fill this gap, in particular extending some results of H. Poppe and of G. Beer to the general case

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Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Wang

Wei

2007

Topology and its Applications

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Hyperspace dynamical system (2 E , 2 f ) induced by a given dynamical system (E, f ) has been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on 2 E employed in these studies is non-metrizable when E is not compact metrizable, e.g., E = R n . Consequently, metric related dynamical concepts of (2 E , 2 f ) such as sensitivity on initial conditions and metric-based entropy, could not even be defined. Moreover, a condition on (2 E , 2 f ) equivalent to the transitivity of (E, f ) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on 2 E is again HLCSC, thus metrizable. In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (E, f ), these three conditions on (E, f ) are respectively equivalent to mixing, weak mixing and transitivity on (2 E , 2 f ) (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.

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Note on hit-and-miss topologies

Cited by 3 publications

References 15 publications

Strong Proximities on Smooth Manifolds and Vorono\" i Diagrams

Strong Proximities on Smooth Manifolds and Vorono\" i Diagrams

Fell topology on the hyperspace of a non-Hausdorff space

Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

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