Infinite Union of Closed Sets of Metric Space may not be Closed

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Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Let $V_1, V_2, V_3, \ldots$ be an infinite set of closed sets of $M$.


Then it is not necessarily the case that $\displaystyle \bigcup_{n \mathop \in \N} V_n$ is itself a closed set of $M$.


Proof

Consider the closed real interval:

$\left[{\dfrac 1 n \,.\,.\, 1}\right] \subseteq \R$

From Closed Real Interval is Closed Set, $\left[{\dfrac 1 n \,.\,.\, 1}\right]$ is closed in $\R$ for all $n \in \N$.

But:

$\displaystyle \bigcup_{n \mathop \in \N} \left[{\dfrac 1 n \,.\,.\, 1}\right] = \left({0 \,.\,.\, 1}\right]$

The result follows from Half-Open Real Interval is neither Open nor Closed.

$\blacksquare$


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