Countable dense homogeneity of definable spaces

Abstract

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of-or-equal-to 2 Superscript omega"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X\subseteq 2^{\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the following are equivalent: (1) <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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>δ</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript omega"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, (2) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript omega"> <mml:semantics> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">X^{\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is CDH and (3) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript omega"> <mml:semantics> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">X^{\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript omega"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Superscript omega"> <mml:semantics> <mml:msup> <mml:mi>ω</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\omega ^{\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mtext>a</mml:mtext> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {\text {a}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>ns and Zhou, by showing that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German p equals min left-brace kappa colon 2 Superscript kappa"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">min</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>κ</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>κ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {p}= \min \{\kappa : 2^{\kappa }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not CDH<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right-brace"> <mml:semantics> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:annotation encoding="application/x-tex">\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.