Let (X,d) be a metric space and consider the subset A ⊂ X and B ⊂ X . Show that the closure satisfies the following property closure( A ∪ B) =closure(A) ∪ closure(B)
First of all, let us remark that the closure of a set is the smallest closed set containing . Indeed, any closed set that contains should also contain its closure by its definition, and the closure is itself a closed set.
Secondly, let us remark that if then . It is just a direct application of the definiton of closure (or we can use the property we just mentioned above).
Finally, let us remind that a finite union of closed sets is itself closed.
From these three properties we deduce easily the result :
is the smallest closed set containing
is a closed set, it contains and B and thus . Therefore,
At the same time, , so we should have . Same argument for gives us that
By the double inclusion we have the result.
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