1987/07/01 by Robert Williamson, Ludvík Janoš · 2 citations
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #Algorithm #Annotation #Artificial intelligence #Biology #Computer science #Functional Equations Stability Results #Type (biology)
paper · pdf · doi:10.1090/s0002-9939-1987-0891165-x
openalex publication_date 1987/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27
A metric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma d right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to be Heine-Borel if any closed and bounded subset of it is compact. We show that any locally compact and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">σ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -compact metric space can be made Heine-Borel by a suitable remetrization. Furthermore we prove that if the original metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is complete, then this can be done so that the new Heine-Borel metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d prime"> <mml:semantics> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">d’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally identical to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e., for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x ∈ X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a neighborhood of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on which the two metrics coincide.