I am unable to visualize complex shapes in space. I can draw simpler shapes and explain which points correspond to which but can't for the life of me determine the name of the shape. I would like somebody to help me identify those. I will present my thought process through a picture since it is very difficult to explain with words.
I am given the quotients $\mathbb{R}/\mathbb{Z} \times[0,1]/\sim$ where $\sim$ denotes $(t,0) \sim (-t,1)$ and $\mathbb{R}/\mathbb{Z} \times[0,1]/\sim'$ where $\sim'$ denotes $(t,0) \sim (t,1)$.
I think the first quotient represents a cylinder whose opposite edges at extremities are "identified", i.e. it looks kind of a Möbius strip but I don't think it is the Möbius strip because of instead of having sides as extremities there are circles. I have made a drawing to see things better but I still can't identify the correct shape. Maybe some kind of 8-looking strip that's not Möbius?
The second one seems easier and I guess it's just the torus... Am I right?
For the first one:
if I try to "identify" the circles in my head I just get some kind of strip that intersects itself like an "8" strip. Does this shape have a name? I have low spatial IQ