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Mar 22, 2017 at 11:59 comment added Tom Goodwillie Assuming these are Hausdorff spaces, the plane must be locally homeomorphic to the product of two of its lines, so it seems that it must be even-dimensional.
Mar 21, 2017 at 23:33 comment added Włodzimierz Holsztyński It's rather clear that the topological projective planes should be topologically homogenous. In the compact case they cannot be 1-dimensional.
Mar 21, 2017 at 12:46 comment added Ali Taghavi Prof. GoodWillie thank you very much for your interesting answer. I am thinking to replacement of other spaces by the interval for example circle, spheres,.etc.
Mar 21, 2017 at 12:28 vote accept Ali Taghavi
Mar 21, 2017 at 1:07 history answered Tom Goodwillie CC BY-SA 3.0