# Quotient space in topology

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.

• The relation is "identify all points in $A$ together, all other points are identified with themselves." – Randall Mar 28 '19 at 13:06
• The most natural IMHO is to describe the set of equivalence classes as $\{\{x\}: x \notin A\} \cup \{A\}$, so we" identify $A$ to a point", as it's called. – Henno Brandsma Mar 28 '19 at 17:14

$$x\sim y\iff x=y\text{ or }x\text{ and }y\text{ are both elements of subset }A$$
• Thanks. So basically the entire subset $A$ becomes one point in the quotient space, and all other points remain the same right? – Sigurd Mar 28 '19 at 13:09