## closed sets in a pre-image

1. The problem statement, all variables and given/known data
This is a topology problem.
I have a continuous map from X to Y, and I take an open set U in Y, and I look at its preimage. Is it true that there must always be an open set in X whose closure is in the preimage of U?

I know that there is always an open set whose closure is in the preimage of the closure of U. But that is not the same thing...
2. Relevant equations

3. The attempt at a solution

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 Recognitions: Homework Help Science Advisor The preimage of an open set in Y IS an open set in X. That's basically the definition of continuous function. If it's closure is also the preimage, then the answer is no. Not every open set is also closed. Why are you muddling this up?
 I am not asking whether the closure of the preimage of an open set is in the preimage, I am asking if there is any open set whose closure is in the preimage.

Recognitions:
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