Show that $F\subset A$ is closed if and only if exists a closed set $W$ such that $F=W\cap A$.
Characterization of closed set
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general-topology
metric-spaces
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2If $A = [0,2)$ and $W = [1,3]$, then $F = W \cap A = [1,2)$ is not closed – 2017-02-26
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0Do you mean closed in the subspace $A$? – 2017-02-26
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2@AndreiKulunchakov $[1,2)$ is closed in the subspace topology on $[0,2)$. – 2017-02-26
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0Y es closed un A – 2017-02-26
1 Answers
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A subset of $A$ is closed if and only if $F \setminus A$ is open, which means that $F \setminus A = A \cap W$ for some open set in the ambient space. Using principles of set theory, $$ F \setminus A = A \cap W \implies F = A \cap W $$ (Draw a venn diagram and convince yourself.)