A metric space with the Haver property whose square fails this property

Pol, Elżbieta; Pol, Roman

doi:10.1090/s0002-9939-08-09511-7

Cited by 11 publications

(15 citation statements)

References 8 publications

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“…Section 6, (A), and our second result confirms (under MA) a conjecture formulated in [ Our constructions combine several ideas from the papers [16,17,20]. An important element in these constructions is the celebrated Michael technique [14] concerning concentrated sets in product spaces.…”

supporting

confidence: 81%

“…The original, more handy definition, is the following one: a space X has the property C if and only if for each sequence [1] and [6] (cf. also [17,Section 4…”

Section: Preliminariesmentioning

confidence: 99%

“…(5). The idea of the proof is taken from [17]. For each m, fix open sets V 0 (m), V 1 (m) in T such that, for s = 0, 1,…”

Section: Michael's Concentrated Setsmentioning

confidence: 99%

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On metric spaces with the Haver property which are Menger spaces

Pol

2010

Topology and its Applications

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A metric space (X, d) has the Haver property if for each sequence 1 , 2 , . . . of positive numbers there exist disjoint open collections V 1 , V 2 , . . . of open subsets of X, with diameters of members of V i less than i and ∞ i=1 V i covering X, and the Menger property is a classical covering counterpart to σ -compactness. We show that, under Martin's Axiom MA, the metric square (X, d) × (X, d) of a separable metric space with the Haver property can fail this property, even if X 2 is a Menger space, and that there is a separable normed linear Menger space M such that (M, d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].

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supporting

confidence: 81%

“…The original, more handy definition, is the following one: a space X has the property C if and only if for each sequence [1] and [6] (cf. also [17,Section 4…”

Section: Preliminariesmentioning

confidence: 99%

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On metric spaces with the Haver property which are Menger spaces

Pol

2010

Topology and its Applications

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“…The corresponding connection for the Haver property and S c (O, O) was commented on by Babinkostova in [4] and was followed up on by E. Pol and R. Pol in [24] where it is stated that A metrizable space X has property C if and only if for any metric d on X generating the topology, (X, d) has the Haver property.…”

Section: Another Relation Between Strong Haver and S D (O O)mentioning

confidence: 98%

“…Pol in comment (D) of [24]. It is a straightforward application of definitions to see the forward direction.…”

Section: Another Relation Between Strong Haver and S D (O O)mentioning

confidence: 99%

Selective Strong Screenability

Coombs¹

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Screenability and strong screenability were both introduced some sixty years ago by R.H. Bing in his paper Metrization of Topological Spaces. Since then, much work has been done in exploring selective screenability (the selective version of screenability). However, the corresponding selective version of strong screenability has been virtually ignored. In this paper we seek to remedy this oversight. It is found that a great deal of the proofs about selective screenability readily carry over to proofs for the analogous version for selective strong screenability. We give some examples of selective strongly screenable spaces with the primary example being Pol's space. We go on to explore a natural weakening of selective strong screenability in topological groups.We conclude with an exploration of the difficulty in extending discrete families of sets as well as giving several directions one might go in when continuing the exploration of selective strong screenability.iv

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