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• W3212970028 abstract "We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [Topology Appl. 14 (1982), pp. 151–161], which was later reposed by A. Miller, T. Orenshtein, and B. Tsaban. Namely, we show that under <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German p equals German c> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>p</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>c</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {p}=mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-set that is not a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-set. Thus we constructed a subset <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of reals such that the space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper C Subscript p Baseline left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathrm {C}_p(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of all real-valued continuous functions on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not Fréchet–Urysohn, but possesses the Pytkeev property. Moreover, under <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper C bold upper H> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>C</mml:mi> <mml:mi mathvariant=bold>H</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {CH}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-set that is not a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-set solving a problem by M. Sakai. In fact, we construct various examples of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-sets that are not <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding=application/x-tex>gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Fréchet–Urysohn property for many different Borel ideals in the realm of functional spaces." @default.
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• W3212970028 date "2023-09-12" @default.
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• W3212970028 title "Ideal approach to convergence in functional spaces" @default.
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• W3212970028 doi "https://doi.org/10.1090/tran/9008" @default.
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