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Finite versus infinite: an insufficient shift

Yann Pequignot

Abstract

The shift graph is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form $G_{f}$ defined on some Polish space $X$, where two distinct points are adjacent if one can be obtained from the other by a given Borel function $f:X\to X$. This answers the primary outstanding question from \cite{Kechris19991}.

Finite versus infinite: an insufficient shift

Abstract

The shift graph is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form defined on some Polish space , where two distinct points are adjacent if one can be obtained from the other by a given Borel function . This answers the primary outstanding question from \cite{Kechris19991}.

Paper Structure

This paper contains 1 section, 2 theorems, 14 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

There exists a Polish space $X$ together with a continuous finite-to-$1$ function $f:X\to X$ such that $\chi_{B}(\mathcal{G}_{f})=\aleph_{0}$ and there is no Borel homomorphism from $\mathcal{G}_{\mathsf{S}}$ to $\mathcal{G}_{f}$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['mainThm']}