All posts by Joanna

Orbit-finite bijections can be smoothed

In his recent post Bartek demonstrated an example of two equivariant sets with atoms that are related by a finitely supported bijection, but not by any equivariant bijection. We show that if the sets are orbit-finite then this situation is impossible.

Proposition. Let X and Y be two orbit-finite equivariant sets with atoms. If there exists a finitely supported bijection f \colon X \rightarrow Y then there exists an equivariant bijection F \colon X \rightarrow Y.
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