I know that the Cantor set is countable dense homogeneous. My question is: if A,B,C,D are countable dense subsets of the Cantor set such that the pairs A and B and C and D are disjoint, there exists a homeomorphism f of the Cantor set such that f(A) = C and f(B) = D?
2026-03-25 12:53:26.1774443206
Is the Cantor set countable dense homogeneous in pairs?
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Yes, it is. The reason is that the Cantor set is strongly locally homegeneous (SLH) and Polish. The same proof by Bessaga and Pelczynski that shows that SLH Polish spaces are countable dense homogeneous gives what you ask. In fact, it is not known whether there is any space which is CDH and not CDH in pairs.