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• W1536023218 abstract "A space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a property <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper P> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>P</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {P}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>strictly</italic> if every finite power of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper P> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>P</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {P}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>A condensation</italic> is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is countable, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be condensed onto a strictly hereditarily separable space. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s left-parenthesis upper C Subscript p Baseline left-parenthesis upper X right-parenthesis right-parenthesis less-than-or-equal-to omega> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ω<!-- ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>s(C_{p}(X))leq omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript p Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{p}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript p Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{p}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly hereditarily separable. Under <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper M upper A plus normal not-sign upper C upper H right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi mathvariant=normal>¬<!-- ¬ --></mml:mi> <mml:mi>C</mml:mi> <mml:mi>H</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(MA+neg CH)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a topological group such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s left-parenthesis upper C Subscript p Baseline left-parenthesis upper G right-parenthesis right-parenthesis less-than-or-equal-to omega> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ω<!-- ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>s(C_{p}(G))leq omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly hereditarily Lindelöf and strictly hereditarily separable." @default.
• W1536023218 created "2016-06-24" @default.
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• W1536023218 date "1996-01-01" @default.
• W1536023218 modified "2023-10-18" @default.
• W1536023218 title "On spread and condensations" @default.
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