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Lets $X$ is a topological space and $Y$ is some subset of $X$. How we define topology of quotient space $$X/Y=\{x\in X~|~x\sim y\Leftrightarrow x, y\in Y\}.$$

Thanks.

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    There's something wrong in the formula. I think you want to collapse $Y$, so the equivalence relation you must consider is defined by: $x \sim x'$ iff $x = x'$ or $x, x' \in Y$. Now consider the quotient projection $p : X \rightarrow X / \sim$ and define the topology of $X/ \sim$ saying that a subset $A \subset X/ \sim$ is open iff $p^{-1}(A)$ is open in $X$.2011-09-11
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    @Andrea: it is another definition. In "my" construction points of $Y$ was identified. (e.g. $D^{n}/\partial D^{n}\thickapprox S^{n}$)2011-09-11
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    I think Andrea's answer still applies. Given a set $A$ in $X/Y$, either it contains a point of $Y$, or it doesn't. If it doesn't, it's open in $X/Y$ iff it's open in $X$. If it does, it's open in $X/Y$ if and only if $S\cup Y$ is open in $X$.2011-09-11

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