Closure is injective on irreducible closed sets + ambiguity in proposition

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

• What is $V$? $Z$? – Hagen von Eitzen Jun 11 at 21:23
• @HagenvonEitzen sorry they are the same – Pedro Jun 11 at 21:25
• Where do you use that $Y$ and $Z$ are closed irreducible? Btw, what does it mean? – Berci Jun 11 at 22:05
• @Berci good point i do not use it. A set is irreducible if it cannot be written as the union of two proper closed subsets of it – Pedro Jun 11 at 22:15
• And your interpretation about closures is also correct. – Berci Jun 11 at 22:23