# A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds.

I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff.

It seems easy to see that the Möbius Strip is locally homeomorphic to $\mathbb{R}^2$ from its plot, however I struggle formalizing the idea, and basically the same can be said about Hausdorff.

The Projective Plane seems Hausdorff because defining it as the quotient of the sphere by the antipodal relation. It appears that it is enough to take to disjoint open sets in the sphere and then apply the quotient map, but I cannot prove it.

• Your idea for the projective plane can be made rigorous. What definition of the Möbius strip are you using? – Michael Albanese Mar 16 '15 at 4:11
• @MichaelAlbanese Mobius Strip is $[0,1]\times(0,1)$ with the equivalence relation $(0,y)\sim(1,1-y)$ – user203327 Mar 18 '15 at 23:34