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I have added statement of the basic fact that -spheres are cosets of orthogonal groups to coset space (also to n-sphere). Then I added a section “Properties – Sequences of coset spaces” with the basic statement about sequences induced from the consecutive inclusion of two subgroups, and an example involving orthogonal groups. Just basic stuff, for reference.
I have changed the redirects: “coset space” used to point to homogeneous space instead of to coset. I changed that and accordingly moved the above additions to the entry coset.
Have added a section Properties – Quotient maps with some sufficient conditions on when coset projections are fiber bundles. I have also copied these over to principal bundle – Properties – coset projections
Why coset space is separated from previous entry homogeneous space ? Is there any real difference ? Well, I remember that one assumes Hausdorff paracompact spaces to have 1-1 correspondence between the two points of view, but has this observation being precisely implemented in this split ? I am just noticing this to make peopkle aware that we need some readjusting eventually.
In noncommutative geometry, in the quantum group case the common definitions like of the quantum homogeneous space and of the quantum coset space are quite different and some specialization is also needed to have an exact correspondence.
Yes, the orignal entry has a subsection (here) discussing the relation. Or rather: pointing to a place where it is discussed.
Formally, homogeneous spaces are more general, the coset spaces are the archetypal examples, that is, under weak assumptions the general examples up to isomorphisms.
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