This may be a very simple question about "written-math" but as a non-english speaker it maker the following proposition difficult to understand
Proposition. Let $X$ be a topological space. If $U$ is an open set of $X$ then the map $Y\mapsto\overline{Y}$, restricted to closed irreducible sets $Y$ in $U$ is injective
The sets $Y$ are $X$-closed or $U$-closed (very different things since $U$ is open)?. Is there an established "semantic" difference between closed sets in $U$ and closed sets of $U$?
I assume the closure is the $X$-closure because there is no indication to do otherwise. Then the sets $Y$ should be $U$-closed for the closure to be a non-trivial map. Then why use $U$ open set of $X$ but $Y$ closed in $U$?
So let $Y,Z\subseteq U$ be closed sets (of/in? $U$) such that $\overline{Y}=\overline{Z}$. Then $Y=Y_1\cap U$ and $Z=Z_1\cap U$ for $Y_1,Z_1$ closed sets of $X$ $$Z\subseteq\overline{Z}=\overline{Y}=\overline{Y_1\cap U}\subseteq Y_1\cap\overline{U}\subseteq Y_1$$ $$Z=Z\cap U\subseteq Y_1\cap U=Y$$ And $Y\subseteq Z$ for the same reason so $Y=Z$. Is this proof correct?
Thanks