Intervals of continua which are Hilbert cubes

Abstract

If <italic>P</italic> is a subcontinuum of a metric continuum <italic>X</italic>, then by the <italic>interval of continua</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper P comma upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(P,X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we mean the space of all subcontinua of <italic>X</italic> which contain <italic>P</italic> (with the Hausdorff metric). We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper P comma upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(P,X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is often homeomorphic with the Hilbert cube.

Locations

• Proceedings of the American Mathematical Society - View - PDF