Homeomorphism groups and homogeneous spaces

van Mill, Jan

doi:10.1090/S0002-9939-1984-0759672-5

Homeomorphism groups and homogeneous spaces
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Proc. Amer. Math. Soc. 92 (1984), 449-454 Request permission

Abstract:

We show that the $2$-sphere ${S^2}$ can be decomposed into two sets $A$ and $B$ such that (1) both $A$ and $B$ are connected, locally connected, dense and Baire, (2) both $A$ and $B$ are (much more than) topologically homogeneous, (3) $A$ is not homeomorphic to $B$, and (4) the autohomeomorphism groups of $A$ and $B$ are (algebraically) isomorphic.

References

D. W. Curtis and J. van Mill, Zero-dimensional countable dense unions of $Z$-sets in the Hilbert cube, Fund. Math. 118 (1983), no. 2, 103–108. MR 732658, DOI 10.4064/fm-118-2-103-108
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
N. J. Fine and G. E. Schweigert, On the group of homeomorphisms of an arc, Ann. of Math. (2) 62 (1955), 237–253. MR 72460, DOI 10.2307/1969678
Murray Gerstenhaber, On canonical constructions. I, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 233–236. MR 74417, DOI 10.1073/pnas.41.4.233
Murray Gerstenhaber, On canonical constructions. II, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 881–883. MR 84499, DOI 10.1073/pnas.42.11.881
Murray Gerstenhaber, On canonical constructions. IV, Proc. Amer. Math. Soc. 8 (1957), 745–749. MR 132787, DOI 10.1090/S0002-9939-1957-0132787-4
Jan van Mill, Strong local homogeneity does not imply countable dense homogeneity, Proc. Amer. Math. Soc. 84 (1982), no. 1, 143–148. MR 633296, DOI 10.1090/S0002-9939-1982-0633296-0
Jan van Mill and Evert Wattel, Partitioning spaces into homeomorphic rigid parts, Colloq. Math. 50 (1985), no. 1, 95–102. MR 818090, DOI 10.4064/cm-50-1-95-102
Matatyahu Rubin, On the reconstruction of topological spaces from their groups of homeomorphisms, Trans. Amer. Math. Soc. 312 (1989), no. 2, 487–538. MR 988881, DOI 10.1090/S0002-9947-1989-0988881-4
Martin T. Wechsler, Homeomorphism groups of certain topological spaces, Ann. of Math. (2) 62 (1955), 360–373. MR 72453, DOI 10.2307/1970069
James V. Whittaker, On isomorphic groups and homeomorphic spaces, Ann. of Math. (2) 78 (1963), 74–91. MR 150750, DOI 10.2307/1970503