Projective versions of selection principles

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.2010 Mathematics Subject Classification. 37F20, 26A03, 03E75, 54C35.

Section: Additional Resultsmentioning

confidence: 99%

Strongly sequentially separable function spaces, via selection principles

Osipov

Szewczak

Tsaban

2020

“…Since (2) states that H is a Rothberger bounded subset of G, Theorem 3 gives the implication from (2) to (3) and from (3) to (5). Also, (3) implies (4) which implies (2).…”

Section: Lemma 2 Let (X U ) Be a Uniform Space And Let K Be A Compamentioning

confidence: 90%

“…Since (2) states that H is a Rothberger bounded subset of G, Theorem 3 gives the implication from (2) to (3) and from (3) to (5). Also, (3) implies (4) which implies (2). The proof that (5) implies (1) uses the fact that [1] in that it does not require the group (G, * ) to be metrizable.…”

Section: Lemma 2 Let (X U ) Be a Uniform Space And Let K Be A Compamentioning

confidence: 97%

Rothberger bounded groups and Ramsey theory

Scheepers

2011

OverviewBoundedness properties in topological groups are counterparts for covering properties in general topological spaces:Guran's notion of ℵ 0 -boundedness is a counterpart of the Lindelöf covering property [9], while Okunev's notion of o-boundedness (named Menger-boundedness by Kočinac, who introduced this notion independently) and Tkachenko's corresponding property of strict o-boundedness are counterparts of σ -compactness [10]. In this paper we consider a boundedness property which approximates Borel's metric notion of strong measure zero. This boundedness property was considered in unpublished work by Galvin, was later independently considered by Kočinac under the name of Rothberger boundedness, the terminology used since [1]. It should be noted, however, that the property now called "Rothberger bounded" was already considered by Rothberger in the Hilfssatz appearing on p. 51 of [16].In [1] it was shown that a subgroup (or subset) of a metrizable topological group is Rothberger bounded if, and only if, it is strong measure zero in all left invariant metrics of the group. In [1] we also extended some of the characterizations of strong measure zero of [19] to Rothberger boundedness, but we did not have techniques to also extend the Ramseytheoretic characterization to this context. In [13] Kočinac further extends notions of boundedness in topological groups to corresponding notions of boundedness in uniform spaces. One objective of our paper is to show how to extend the Ramseytheoretic characterization of the metric notion of strong measure zero to a Ramsey-theoretic characterization of Rothberger

“…Projectively countable Lindelöf spaces are Rothberger [19], [5], [26]; in fact Proposition 13 [22]. Borel's Conjecture is equivalent to the assertion that a space is Rothberger if and only if it is projectively countable.…”

Section: Lemma 12 Perfect Maps Preserve Countable Typementioning

confidence: 99%

“…Proof. Projectively countable Lindelöf spaces are Rothberger [5], [19], [26] and hence indestructible. By the ℵ 1 -Borel Conjecture, they are then projectively ℵ 1 .…”

Section: Lemma 12 Perfect Maps Preserve Countable Typementioning

confidence: 99%

Productively Lindelöf and indestructibly Lindelöf spaces

Duanmu

Tall

Zdomskyy

2013

There has recently been considerable interest in productively Lindelöf spaces, i.e. spaces such that their product with every Lindelöf space is Lindelöf. See e.g.[4], [29], [1], [26], and work in progress by Miller, Tsaban, and Zdomskyy, Repovs and Zdomskyy, and by Brendle and Raghavan. Here we make several related remarks about such spaces. Indestructible Lindelöf spaces, i.e. spaces that remain Lindelöf in every countably closed forcing extension, were introduced in [27]. Their connection with topological games and selection principles was explored in [25]. We find further connections here.