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Suppose $X,τ$ a topological space. If $A$ and $B$ are any two subsets of $X$ prove or disprove that $CL(A∩B)=CL(A)∩CL(B)$

I know that closed sets is closed under intersection, however I still got the feeling that this is not true. I wonder if anyone have an counter example.

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I wonder whether this should be closed as a duplicate of this question:… (The only difference is that the other question asks about metric spaces. Of course, any counterexample which works in a metric space is also a counterexample showing that the equality is not true in topological spaces.) – Martin Sleziak Feb 4 '14 at 14:23
up vote 7 down vote accepted

Try $A=[0,1)$ and $B=(1,2]$ as a counterexample.

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