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.

  • $\begingroup$ The relation is "identify all points in $A$ together, all other points are identified with themselves." $\endgroup$ – Randall Mar 28 '19 at 13:06
  • $\begingroup$ 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. $\endgroup$ – Henno Brandsma Mar 28 '19 at 17:14

$$x\sim y\iff x=y\text{ or }x\text{ and }y\text{ are both elements of subset }A$$

| cite | improve this answer | |
  • $\begingroup$ Thanks. So basically the entire subset $A$ becomes one point in the quotient space, and all other points remain the same right? $\endgroup$ – Sigurd Mar 28 '19 at 13:09
  • $\begingroup$ That is correct. $\endgroup$ – drhab Mar 28 '19 at 13:10
  • $\begingroup$ But you must also get the topology right. $\endgroup$ – Randall Mar 28 '19 at 17:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.