# Thread: Path in S2 empty interior

1. ## 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

2. 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 $\mathbb{R}^2$ and we're talking here of a homeomorphism (which is topological equivalence), then ANY homeomorphic image of I in $\mathbb{R}^2$ will have the same topological characteristics as I itself...

Tonio

3. 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 $\mathbb{R}^2$ 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.