Fine Topology Methods in Real Analysis and Potential Theory

Lukeš, Jaroslav; Malý, Jan; Zaj́ıček, Luděk

doi:10.1007/bfb0075894

Cited by 115 publications

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“…Thus in the context of Proposition 6.37 and Theorem 6.39 from [10] we have There are two fundamental examples in which, by Theorem 1.16, we get the abstract density topologies. Example 1.18.…”

Section: For Every S-measurablementioning

confidence: 90%

On Density Topologies with Respect to Invariant σ-Ideals

Hejduk¹

2002

Journal of Applied Analysis

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Abstract. The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ-ideals on R. The properties of such topologies, including the separation axioms, are studied. NotationBy R we shall denote the set of all reals numbers and by N the set of positive integers. Let l stand for Lebesgue measure. The capitals L and L denote the σ-algebra of all Lebesgue measurable sets in R and the σ-ideal of all Lebesgue null sets. The natural topology on R is denoted by T 0 . If T is a topology on R, then we fix the notation: B(T ) -the σ-algebra of all Borel sets with respect to T , Ba(T ) -the σ-algebra of all sets having the Baire property with respect to T , K(T ) -the σ-ideal of all meager sets with respect to T .2000 Mathematics Subject Classification. 28A05, 54A10. Key words and phrases. Density point, density topology, the separation axioms, invariant ideals and algebras.

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Section: For Every S-measurablementioning

confidence: 90%

On Density Topologies with Respect to Invariant σ-Ideals

Hejduk¹

2002

Journal of Applied Analysis

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“…[4]). In particular, if V ∈ τ 0 then V = W \ P where W is regular open and P is a nowhere dense closed set (see [5]).…”

Section: Introductionmentioning

confidence: 99%

“…Since the pair (Ba, K) has the hull property, what means that every family of pairwise disjont sets having the Baire property but not meager is at most countable, and Φ r is a lower density operator on (R, Ba, K), we infer that the family T Φ r = {A ∈ Ba; A ⊂ Φ r (A)} is a topology on R, called an abstract density topology on (R, Ba, K) (see [4], p. 208 and p. 213). If V ∈ τ 0 then by Remark 5.1, V = W \ P where W is a regular open set and P ∈ K. Hence G(A) = W and Φ r (V ) = Φ(W ) ⊃ W ⊃ V .…”

Section: Introductionmentioning

confidence: 99%

On equivalence of topological and restrictional continuity

Flak¹,

Hejduk²

2015

Modern Real Analysis

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Let R denote the set of reals and N the set of positive integers. By τ 0 we shall denote the natural topology on R. Let B(τ), K(τ), Ba(τ) denote the family of all Borel sets, meager sets and sets having the Baire property in a topologicalwhere int τ and cl τ mean the interior and closure with respect to the topology τ. If τ = τ 0 then we shall use the notation B, K and Ba, respectively. The symmetric difference of sets A, B is denoted by A B.Let Φ : τ 0 → 2 R be an operator satisfying the following conditions:Let Φ stand for the family for all operators satisfying conditions (i) − (iii).Remark 5.1. If Φ ∈ Φ then Φ(A) ⊂ cl τ 0 A for every A ∈ τ 0 .

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“…Ì ÓÖ Ñ 3 (cf. [7])º A topology τ on X is the abstract density topology on (X, S, J ) if and only if the following conditions are satisfied:…”

mentioning

confidence: 99%

“…[7])º If Φ : S → S is the lower density operator on (X, S, J ) and the pair (S, J ) has the hull property then the family…”

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confidence: 99%

One more difference between measure and category

Hejduk¹

2011

Tatra Mountains Mathematical Publications

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ABSTRACT. The first part of the paper contains some ideas of the density topologies in the measurable spaces. The second part is devoted to the difference between measure and category for the abstract density space related to the separation axioms.Let X be a nonempty set, A ⊂ 2 X be an arbitrary family of sets, and Φ be an operator such thatIf the family T Φ = A ∈ A : A ⊂ Φ(A) is a topology, then we say that the topology T Φ is generated by the operator Φ.Let (X, S, J ) be a measurable space, where S is the σ-algebra and J ⊂ S is the proper σ-ideal. The fact that for any sets A, B ∈ S we have A△B ∈ J will be denoted by A ∼ B.Ò Ø ÓÒ 1º We shall say that a topology τ on X is the abstract density topology on (X, S, J ), if there exists a lower density operator Φ : S → S such that T Φ = τ. Ò Ø ÓÒ 2º An operator Φ : S → S is called the lower density operator on (X, S, J ) if:The following states when a topology τ is the abstract density topology.

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Fine Topology Methods in Real Analysis and Potential Theory

Cited by 115 publications

References 0 publications

On Density Topologies with Respect to Invariant σ-Ideals

On Density Topologies with Respect to Invariant σ-Ideals

On equivalence of topological and restrictional continuity

One more difference between measure and category

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