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.
You can use your technique with the projective plane for the Mobius strip as well.
When you quotient by the antipodal map on the 2-sphere, a symmetrical cylindrical strip around the equator projects to a Mobius strip inside the projective plane. Antipodal neighborhoods in the equatorial cylinder project to single neighborhoods in the Mobius strip.