Give an example of a function $f:X \to Y$ and a subset $A \subset X$ such that f is open but $f_A$, the restriction of $f$ to $A$ is not open.

Can someone help me please? Thanks

  • $\begingroup$ what do you mean by f is open? $\endgroup$ – RowanS Jun 21 '15 at 19:40
  • $\begingroup$ @Rowan: It’s a standard notion: it means that if $U$ is open in the domain of $f$, then $f[U]$ is open in the codomain. $\endgroup$ – Brian M. Scott Jun 21 '15 at 19:44

HINT: Let $X=Y=\Bbb R$, take $f$ to be the identity function, and take $A$ to be a set that isn’t open in $X$.

  • $\begingroup$ Thanks, the example was easy to find $\endgroup$ – Ryoma Jun 21 '15 at 19:53
  • $\begingroup$ @Ryoma: You’re welcome. $\endgroup$ – Brian M. Scott Jun 21 '15 at 19:55
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    $\begingroup$ @Ryoma: If you feel that this answer has satisfied your question, it is encouraged that you accept the answer. Since you haven't accepted any answers before (as of now), I thought I'd point out this link for your information. $\endgroup$ – Prism Jun 22 '15 at 5:06

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