I found quotient space on https://en.wikipedia.org/wiki/Quotient_space, I had seen a property that I don't know how I can prove it:
$X/\sim$ is a $T_1$ space if and only if every equivalence class of $\sim$ is closed in $X$.
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A topological space $\,Y\,$ is $\;T_1\;$ iff every singleton $\,\{y\}\;,\;y\in Y\;$ , is closed (yes, I know the usual definition is another one. This is an equivalent one).
Well, what are the points in the quotient space $\,X/\sim\;$ and how's the topology on this space defined...?
DonAntonio
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$[x]$ is closed in $X/\sim$ by its Hausdorff. If $p$ is quotient map. Then $[x]$ is closed in $X/\sim$ iff $p^{-1}([x])$ is closed in $X$ – Muniain Jun 21 '13 at 16:37
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but it isn't obvious – Muniain Jun 21 '13 at 16:39
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I'm not sure what "isn't obvious", @Muniain, but it doesn't matter: yes, in general things must be proved. – DonAntonio Jun 21 '13 at 17:08
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Your first comment: what/who is Hausdorff? I can't see anywhere such an assumption, and even if $,X,$ is Hausdorff: how from there can you conclude an equivalence class is closed? – DonAntonio Jun 21 '13 at 17:15
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oh, sorry. From $X/\sim$ is $T_1$ iff $[x]$ is closed in $X/\sim$, then $p^{-1}([x])$ is closed in $X$ – Muniain Jun 22 '13 at 01:28
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Can you give me a proof or material that I can read? – Muniain Jun 22 '13 at 01:32