Understanding the construction for the topological cone $c(S^1) = (S^1 \times [0,1]) \ / \ (S^1 \times \{1\})$. 
I have trouble understanding the construction for the topological cone $c(S^1) = (S^1 \times [0,1]) \ / \ (S^1 \times \{1\})$.

Isn’t $(S^1 \times \{1\})$ just the unit sphere lifted along the $z-$axis by constant height of $1$? I get that this is supposed to represent the vertex of the cone, but don’t understand how? Also what are the equivalence classes for this quotient space? This is very confusing, it seems that the notation $X/A$ is used indistinguishably for quotient spaces and when you deform a set to a point? Could someone help me with understanding this properly?
 A: You start by considering the cylinder $S^1 \times [0,1]$. Now you ask yourself how you would write a horizontal slice of this cylinder. For example, if you slice the cylinder in half, you will have done so along the line $S^1 \times \{\frac12\}$. So indeed, you can think of $S^1 \times \{1\}$ as the unit circle at height 1 above the $xy$-plane. By taking the quotient over $S^1 \times \{1\}$, you're saying that you want to identify all points in $S^1 \times \{1\}$ with each other, i.e. you're treating all points in $S^1 \times \{1\}$ as one point. Since this is now one point, you can think of this point as the vertex of the cone. Let $\sim$ denote your equivalence relation
$$
x \sim y :\iff x,y \in S^1 \times \{1\}
$$
Then, the equivalence classes are the singleton sets $[x]$ if $x \in S^1 \times [0,1)$ and $[(1,1)] = \{ x \in S^1 \times [0,1] \mid x \sim (1,1) \} = S^1 \times \{1\}$, since $(1,1) \in S^1 \times \{1\}$ is a representative of this class. $[x]$ is a singleton if $x \in S^1 \times [0,1)$, because the equivalence relation $\sim$ does not act on these points in any way, so you can think of $[x]$ as just being equal to $x$ itself. I hope this was helpful.
