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Let $(X,d)$ be a metric space. Let $I$ be an infinite set such that for all $i\in I$ let $A_i\subset X$. Show that $$\cup_{i\in I}\bar A_i\subsetneq \overline{\cup_{i\in I} A_i}$$ where the bar denotes closure.

Earlier we showed for just two sets $A,B\subset X$ that $$\bar A \cup \bar B=\overline {A\cup B}$$ so I don't understand how switching to an infinite union breaks this equality, or generally how to prove this statement.

Thanks!

Asaf Karagila
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Yoni
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1 Answers1

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Suppose $X$ is the reals with the standard metric. If $A_i$ is an interval with a lower limit of $1/i$, then $0$ is not in the union of the closures but is in the closure of the union.

Look closely at the wording of the problem as it was stated to you. You probably just need to find a counterexample to show the two sides are not equal.

Browning
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