Some applications of the topological characterizations of the sigma-compact spaces 𝑙²_{𝑓} and Σ

Abstract

We use a technique involving skeletoids in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-compact metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ARs"> <mml:semantics> <mml:mtext>ARs</mml:mtext> <mml:annotation encoding="application/x-tex">\text {ARs}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to obtain some new examples of spaces homeomorphic to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-compact linear spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript f Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">l_f^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For example, we show that (1) every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional metric linear space is homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript f Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">l_f^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; (2) every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-compact metric linear space which is an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="AR"> <mml:semantics> <mml:mtext>AR</mml:mtext> <mml:annotation encoding="application/x-tex">\text {AR}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and which contains an infinite-dimensional compact convex subset is homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and (3) every weak product of a sequence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-compact metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ARs"> <mml:semantics> <mml:mtext>ARs</mml:mtext> <mml:annotation encoding="application/x-tex">\text {ARs}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which contain Hilbert cubes is homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.