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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.

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

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  • $\begingroup$ 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

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