Thread: Path in S2 empty interior

Hi,
What is a good way to prove that the homeomorphic image I of the unit interval in S2 (unit sphere) has empty interior? I can see that any open ball does not lie in I, but do not know how to prove it.

Any topology results can be assumed (it's for a measure theory course) so any suggestions are welcome!

Thanks

Originally Posted by james123

Hi,
What is a good way to prove that the homeomorphic image I of the unit interval in S2 (unit sphere) has empty interior? I can see that any open ball does not lie in I, but do not know how to prove it.

Any topology results can be assumed (it's for a measure theory course) so any suggestions are welcome!

Thanks

I don't know it the following suffices, but since the unit interval I has empty interior in and we're talking here of a homeomorphism (which is topological equivalence), then ANY homeomorphic image of I in will have the same topological characteristics as I itself...

Tonio

Originally Posted by james123

Hi,
What is a good way to prove that the homeomorphic image I of the unit interval in S2 (unit sphere) has empty interior? I can see that any open ball does not lie in I, but do not know how to prove it.

Any topology results can be assumed (it's for a measure theory course) so any suggestions are welcome!

The unit interval had only two cut-points, namely the endpoints of the interval. (A cut-point in a connected space is one whose removal disconnects the space.)

An interior point of a connected set in is never a cut-point. So if the interior is nonempty then there are uncountably many cut points.

To complete the proof, the number of cut-points is invariant under homeomorphism.