Weak orderability of second countable spaces

Gutev, Valentin

doi:10.4064/fm196-3-4

Cited by 17 publications

(3 citation statements)

References 6 publications

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“…non-open component of X, then two points of C, including all noncut points, belong to Z. Such sets and some slight modifications of them were called Purisch sets in [5,14]. One of the best properties of Purisch sets is that they are closed in X, moreover every two such sets are homeomorphic.…”

Section: Suborderable Spaces and Componentsmentioning

confidence: 99%

Weak Selections and Suborderable Metrizable Spaces

Gutev¹

2020

Preprint

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Each continuous weak selection for a space X defines a coarser topology on X, called a selection topology. Spaces whose topology is determined by a collection of such selection topologies are called continuous weak selection spaces. For such spaces, García-Ferreira, Miyazaki, Nogura and Tomita considered the minimal number cws(X) of selection topologies which generate the original topology of X, and called it the cws-number of X. In this paper, we show that cws(X) ≤ 2 for every semi-orderable space X, and that cws(X) = 2 precisely when such a space X has two components and is not orderable. Complementary to this result, we also show that cws(X) = 1 for each suborderable metrizable space X which has at least 3 components.

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Section: Suborderable Spaces and Componentsmentioning

confidence: 99%

Weak Selections and Suborderable Metrizable Spaces

Gutev¹

2020

Preprint

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“…If a connected space X has a continuous weak selection s : ½X 2 ! X , then jnctðX Þj a 2 and ctðX Þ is open and connected [11] (see also [5,Corollary 2.7]). The theorem below extends this property for all n b 2.…”

Section: Cut and Noncut Pointsmentioning

confidence: 99%

Selections and deleted symmetric products

Buhagiar¹,

Gutev²

2017

Tsukuba J. Math.

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We give a very simple example of a connected second countable space X whose hyperspace ½X nþ1 of unordered ðn þ 1Þtuples of points has a continuous selection, but ½X n has none.This settles an open question posed by Michael Hrus ˇa ´k and Ivan Martı ´nez-Ruiz. The substantial part of the paper sheds some light on this phenomenon by showing that in the presence of connectedness this is essentially the only possible example of such spaces.

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“…This fact immediately leads to the following corollary. Preorderable topologies were characterized in Campión et al [6] completing the panorama on orderability of topologies (see Van Dalen and Wattel [36], Purisch [31] or Gutev [24]).…”

Section: The Continuous Representability Property On Topological Spacesmentioning

confidence: 99%