# Let be X a topological space and we suppose that E and F are homeomorphic though a

Let be $X$ a topological space and we suppose that $E$ and $F$ are homeomorphic though a map $f$ from $F$ to $E$. So I ask to me if $E$ is open/closed when $F$ is open/closed but unfortunatley I was not able to prove or to disprove this so that I thought to put a specific question where I ask some clarification: in particular if the result is generally false I'd like to know if it can be true we additional hypotesis, e.g. Hausdorff separability, First Countability, Metric Topology, etc....
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thatuglygirlyu
Consider $E=\left(0,1\right)$ and $F=\mathbb{R}$.
Obviously, $E,F\subset \mathbb{R}$. It is quite easy to show that $F$ and $E$ are homeomorphic.
$F$ is closed and open. But $E$ is not closed.