On Differences of Semi-Continuous Functions

Chaatit, Fouad; Rosenthal, Haskell P.

doi:10.2989/16073600009485979

Cited by 4 publications

(2 citation statements)

References 6 publications

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“…Such maps were introduced by Vinokurov [30] and studied in details in [4,5,6,7,8,11,12,13,14,22,23]. Also they appear naturally in Analysis, see [9,16,19].…”

Section: Introductionmentioning

confidence: 99%

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

Банах

Bokalo

Kolos

2017

Topology and its Applications

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A map f : X → Y between topological spaces is called weakly discontinuous if each subspace A ⊂ X contains an open dense subspace U ⊂ A such that the restriction f |U is continuous. A bijective map f : X → Y between topological spaces is called a weak homeomorphism if f and f −1 are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindelöf number, dimension. Also we classify infinite zero-dimensional σ-Polish metrizable spaces up to a weak homeomorphism and prove that any such space X is weakly homeomorphic to one of 9 spaces: ω, 2 ω , N ω , Q, Q ⊕ 2 ω , Q × 2 ω , Q ⊕ N ω , (Q × 2 ω ) ⊕ N ω , Q × N ω .

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“…Such maps were introduced by Vinokurov [30] and studied in details in [4,5,6,7,8,11,12,13,14,22,23]. Also they appear naturally in Analysis, see [9,16,19].…”

Section: Introductionmentioning

confidence: 99%

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

Банах

Bokalo

Kolos

2017

Topology and its Applications

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“…Real-valued functions of the first stable Baire class on a topological space X naturally appear both as an interesting subclass of all differences of semi-continuous functions on X [6,7] and in problems on the Baire classification of integrals depending on a parameter [2] as well as in problems concerning a composition of Baire functions [8].…”

Section: Introduction Terminology and Notationsmentioning

confidence: 99%

On stable Baire classes

Karlova

Mykhaylyuk

2016

Acta Math. Hungar.

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We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps f : X → Y of the class α for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map f : X → Y belongs to the n'th stable Baire class if and only if there exist a sequence (f k ) ∞ k=1 of continuous maps f k : X → Y and a sequence (F k ) ∞ k=1 of functionally ambiguous sets of the n'th class in X such that f | F k = f k | F k for every k. Moreover, we show that every monotone function f : R → R is of the α'th stable Baire class if and only if it belongs to the first stable Baire class.1991 Mathematics Subject Classification. Primary 54C08, 54H05; Secondary 26A21.

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On scatteredly continuous maps between topological spaces

Банах

Bokalo

2010

Topology and its Applications

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A map f : X → Y between topological spaces is defined to be scatteredly continuous if for each subspace A ⊂ X the restriction f |A has a point of continuity. We show that for a function f : X → Y from a perfectly paracompact hereditarily Baire Preiss-Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset A ⊂ X the set D(f |A) of discontinuity points of f |A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G δ -measurability (the preimage of each open set is of type G δ ). Also under Martin Axiom, we construct a G δ -measurable map f : X → Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov.1991 Mathematics Subject Classification. 54C08.

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On Differences of Semi-Continuous Functions

Cited by 4 publications

References 6 publications

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

On stable Baire classes

On scatteredly continuous maps between topological spaces

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