Characterizing P-spaces X in Terms of C p (X)
Abstract
<jats:p> Dual weak barrelledness led us to prove that <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">P</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -space if and only if every pointwise eventually zero sequence in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is summable, and other better known characterizations. Novel ones recall utility functions from economics and Arkhangel'skii's (strict) <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\tau</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -continuity. Mackey <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\aleph_0</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -barrelled duality leads us to prove that <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is discrete if and only if every bounded <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -compact set in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is relatively compact. We relax the <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -compact hypothesis of Velichko and the <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -countably compact hypothesis of Tkachuk/Shakhmatov to prove : X is a P-space if and only if <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -relatively sequentially complete. </jats:p>