Fixed point sets of metric and nonmetric spaces (1984)

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> is said to have the complete invariance property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis CIP right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mtext>CIP</mml:mtext> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\text {CIP})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if every nonempty closed subset of <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> is the fixed point set of some self-mapping of <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>. It is shown that connected subgroups of the plane and compact groups need not have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CIP"> <mml:semantics> <mml:mtext>CIP</mml:mtext> <mml:annotation encoding="application/x-tex">\text {CIP}</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="CIP"> <mml:semantics> <mml:mtext>CIP</mml:mtext> <mml:annotation encoding="application/x-tex">\text {CIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need not be preserved by self-products of Peano continua, nonmetric manifolds or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CIP"> <mml:semantics> <mml:mtext>CIP</mml:mtext> <mml:annotation encoding="application/x-tex">\text {CIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, an uncountable product of real lines, circles or two-point spaces has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CIP"> <mml:semantics> <mml:mtext>CIP</mml:mtext> <mml:annotation encoding="application/x-tex">\text {CIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Examples are given which contrast the behavior of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CIP"> <mml:semantics> <mml:mtext>CIP</mml:mtext> <mml:annotation encoding="application/x-tex">\text {CIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the nonmetric and metric cases, and examples of spaces are given where the existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CIP"> <mml:semantics> <mml:mtext>CIP</mml:mtext> <mml:annotation encoding="application/x-tex">\text {CIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is neither provable nor refutable with the usual axioms of set theory.