2019/04/22 by Brian, Will
#FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · doi:10.48550/arxiv.1904.09907
The shift map σ is the self-homeomorphism of ω^* = βω∖ ω induced by the successor function n ↦ n+1 on ω. We prove that the isomorphism classes of σ and σ-1 cannot be separated by a Borel set in \mathcal H(ω^*), the space of all self-homeomorphisms of ω^* equipped with the compact-open topology. Van Douwen proved it is consistent for σ and σ-1 not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while σ and σ-1 may not be isomorphic, there is no simple topological property that distinguishes them. As a relatively straightforward consequence of the main theorem, we deduce that OCA+MA implies the set of continuous images of σ fails to be Borel in \mathcal H(ω^*). (Here a ``continuous image'' of σ is meant in the sense of topological dynamics: any h ∈ \mathcal H(ω^*) such that q ∘ σ= h ∘ q for some continuous surjection q: ω^* → ω^*.) This contrasts starkly with a recent theorem of the author showing that under CH, the continuous images of σ form a closed subset of \mathcal H(ω^*).