Let $S^1: x^2+y^2=1$ be the circle ; $S^2: x^2+y^2+z^2=1$ be the sphere; then define also $X= S^1 \times S^2$
- I want to define an equivalence relation on $X$ such that $X/\sim$ is not a top manifold.
- I want to define a non trivial equivalence relation on $X$ such that $X/\sim$ is a top manifold.
I think I can picture $X$ as a sphere revolving around some centre following an orbit which is given by $S^1.$ In other words what I have is a family of spheres parametrized by $S^1.$
For (1), I can identify $E\times (\cos0 , \sin0) \sim E \times (\cos \pi, \sin\pi)$ where $E$ is some point on the equator of the sphere. So that the sphere is moving on an 8 figure, but on the connecting point of the 8 i get two spheres glued by the point $E.$ I think now a neighborhood of $(E,(\cos0 , \sin0))$ will have two connected components, hence this space is not locally euclidean.
For (2), if I identify all the points $$S^2 \times {(\cos 0,\sin0)} \sim S^2 \times (\cos \pi/2, \sin/2 \pi) \sim S^2 \times (\cos \pi, \sin \pi) \sim S^2 \times (\cos 3/2\pi, \sin 3/2\pi) \sim S^2 \times (\cos 2 \pi, \sin 2\pi)$$ what I get is a sphere parametrized by an interval, which is homeomorphic to $S^1 \times S^2$ and hence is a topological manifold.
I would like to know first of all if my reasonings make sense and only secondly some alternative simpler/ better solutions.