I have a short question regarding the following:
$X = S^1 \times \{ 0,1 \} = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, z \in \{ 0,1 \} \} \subset \mathbb{R}^3$, $B_2 = X /_{\sim}, (1,0,0) \sim (1,0,1)$.
Which two points do I have to remove, s.t. $B_2$ and $S^1$ do not have the same number of connected components?
You have two disjoint copies of $S^1$ in space, glued together by identifying one point in each copy.
$B^2$ has one point $q$ (the identified point, really a class of two points) such that $B\setminus \{q\}$ is disconnected (two circles both minus a point, are left).
For $S^1$ (the standard unit circle) it holds that whatever point $p$ we remove from it, $S^1 \setminus \{p\}$ is connected (basically a bent open interval).
This shows that $B^2$ and $S^1$ cannot be homeomorphic, as the former has a cutpoint (as $q$ is named) and $S^1$ has none.