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If $X$ is a topological space and $Y$ a subspace I have the following question.

If $A\subseteq Y$ do we have $$cl_Y(A)=cl_X(A)\cap Y,$$ where $cl_Y$ denotes the closure in the space $Y$?

Try: We have that $cl_X(A)\cap Y$ is a closed subset of $Y$ containing $A$, so this gives $cl_Y(A)\subseteq cl_X(A)\cap Y$. What about the other direction?

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Let $F$ be a closed subset containing $A$, $F= F'\cap X$ where $F'$ is a closed subset of $X$. $F'$ contains $cl_X(A)$. Thus $F'\cap Y= F$ contains $cl_X(A)\cap Y$ which thus the smallest closed subset of $Y$ containing $A$ by definition, it is the adherence of $A$ in $Y$.

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  • $\begingroup$ Don't you mean $F=F'\cap Y$? $\endgroup$ – sqtrat Apr 5 '16 at 13:25

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