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If $X$ is a topological space and $\sim$ is an equivalence relation on $X$, I want to show that $\tau: X \times I \to (X/{\sim}) \times I, \; (x,t) \mapsto (\pi(x),t)$ (where $I =[0,1]$ and $\pi: X \to X/{\sim}$ is the quotient map) is an identification. Obviously, $\tau$ is continuous. But how do I prove that $\tau^{-1}(U)$ is open iff $U$ is open? So far all of my attempts have failed since $\pi$ can't be assumed to be an open map.

Thanks for help in advance.

Michael Hardy
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Raphael Floris
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  • The definition of the topology on the quotient space is ${U|\pi^{-1}(U)$ is open$}$. So $\pi$ is open from definition. – Shaked Dec 12 '17 at 23:10
  • @ShakedBader not all quotient maps are open – Mr. Cooperman Dec 12 '17 at 23:14
  • As I recall this is kind of delicate. Its theorem 20.1 in munkres "elements of algebraic topology." If you dont have that book I can try to give you a screenshot of the proof. It holds more generally if $I$ is replaced with a locally compact hausdorff space. – Mr. Cooperman Dec 12 '17 at 23:18
  • @Tim kinsella Thank you. I shall have a look at it later! – Raphael Floris Dec 12 '17 at 23:26
  • @ShakedBader See https://math.stackexchange.com/questions/655797/example-of-quotient-mapping-that-is-not-open – Raphael Floris Dec 12 '17 at 23:28
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    Note the typographical difference between $X/\sim$ and $X/{\sim}.$ Binary operation symbols have a certain amount of space before and after them (present in $5+3$ but absent in $+3$ or in $5+$) but in $X/{\sim}$ that space shouldn't be there because the symbol isn't being used in that way. But when nothing is before or after it, then those spaces are not there. In other words, code this as X/{\sim}. $\qquad$ – Michael Hardy Dec 13 '17 at 05:40
  • What is an identification? Is it a synonym for a quotient map? – Peter Elias Dec 13 '17 at 07:18
  • @Peter Elias Essentially yes. A continous surjection $f: X \to Y$ is an identification if $U \subseteq Y$ is open iff $f^{-1}(U)$ is open. – Raphael Floris Dec 13 '17 at 07:29

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