I got stuck on this problem about describing quotient space. Hope someone can help me to solve this. Thanks a lot
Let $X$ be topology space and $A \subset X$.Let $\mathcal{P}$ be the partition of $X$ which consists of the set $A$ and the one-point sets $\{x\}$ for all $x \in X - A$. Let $X \backslash A$ be the quotient space with respect to this partition. Describe $X \backslash A$, here:
a) $X = \{(x,y,z)\;|\;x^2 + y^2 = 1\}$ and $A = \{(x,y,0)\;|\;x^2 + y^2 = 1\}$
b) $X = S^2$ and $A$ is the equator in $xy$-place
I got confused when people try to describe quotient space using the intuition, like "imagining...". In my understanding, to describe a quotient space, we need to find a quotient mapping from the original space. So can we describe the quotient space using imagination without clarifying quotient mapping? Is there any chance that our intuition is wrong? Thanks for reading my question. I really appreciate.
$a)$ For the sake of simplicity, cut the cylinder by above and by below in order to get a compact cylinder. Since the equivalence doesn't mess with those parts, it is obvious (prove it) that if we restrict ourselves to this part of the cylinder there will be no problem.
Now take the obvious continuous function from the cylinder to the "two-fold" cone with vertix on the origin. You get a induced continuous map from the quotient space to the cone. Since it is bijective and the cylinder we took is compact (! this is why we restricted. Compactness is good), it is a homeomorphism.
$b)$ Try to repeat the same reasoning as above (it is easier, we are already compact!). Hint: You should arrive at a homeomorphism with two spheres touching at a point.