topology...open map



hello....doctor~

f : (0, 00) -> [-1, 1], f(x) = sin(1/x)

i think that maybe, f(x) is not open map.
so, i must find the open set U such that f(U) is closed set.
but i can't find the example.

so, i need your advice.
thank you very much for your advice.


.



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