On quasi-metric and metric spaces

Abstract

Given 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> with a quasi-metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> it is known that the so-called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -chain approach can be used to produce a metric in <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> equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Superscript p"> <mml:semantics> <mml:msup> <mml:mi> ρ </mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\rho ^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than p less-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0&gt;p\le 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , hence also a quasi-metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde {\rho }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with better properties. We refine this result and obtain an exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is, in general, optimal.