it seems to me sometimes reasonable to distinct 0(d)-point from 3-point or 4-point, in other words, to embed information about embedding space of the geometrical object in question into its name.
so for exemple, a 2-point (a point in E2) is different from a point in E3 - a 3-point. one allows a 2-star of rays, a 2d-star of directions to pass through it, other allows/is open to an axis of planar stars - a 3-star of rays.
so some lines can be more "open" than others, according to where they dwell.
but then, considering the non-orientable 2-point within mobius band, it seems inadequate to catch the nuances of different points. or maybe ratios should be used for non-orientable ones ?
anyway, my question is other but close:
in 4-space, flat this time, i have a patch of 2-plane ( which is actually 4-plane, according to the above)
extending it infinitely, in E4, what do we get ? just ordinary E2 ?
now more important to me (bo would say essential, and he's right this time :wink: ) is how this would behave in S4, or elliptic 4-space, and especially with curvature equal to 1.
would the patch close on itself ? if so then what manifold would we get ?
for 3-space, extending a 3-plane (2-plane in S3), is it projective plane that we get for any initially locally flat patch, when we get to including infinity ?
i wonder if the elliptic 4-plane gives a klein bottle or RP3. ?
isn't every totality of 4-plane(=2-plane spread infinitely in 4-space, infinity excluded, a mobius band ? (i think mobius band is a projective plane minus a point (=with a hole) ?
then the answer to the quaestion at the beginning of this paragraph would be RP3, rather than klein's bottle, it seems to me. :?