In this thread:
What are the epimorphisms in the category of Hausdorff spaces?
the quotient space of a topological space $X$ by a closed set $A \subset X$ is mentioned : $X / A$. This confused me, because I thought you can only quotient by an equivalence relation. In groups, I know that this equivalence relation for $G/H$ is implicitly given by $a \sim b$ iff $ab^{-1} \in H$. But in topological spaces we don't have products and inverses, so my question is, what does this space mean? In particular, can someone tell me explicitly when $x \sim y$ (or $\overline{x} = \overline{y}$) in the space $X/A$? The only thing I can think of is:
$$
\overline{x} = \overline{y} \ \Leftrightarrow x \in A, y \in A,
$$
but I'm not sure if this is correct, it seems like a strange definition to me.
Best Answer
$$x\sim y\iff x=y\text{ or }x\text{ and }y\text{ are both elements of subset }A$$