Infinite Intersection of Open Sets of Metric Space may not be Open

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Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $U_1, U_2, U_3, \ldots$ be an infinite set of open sets of $M$.


Then it is not necessarily the case that $\displaystyle \bigcap_{n \mathop \in \N} U_n$ is itself an open set of $M$.


Proof

Consider the open real interval $\openint {-\dfrac 1 n} {\dfrac 1 n} \subseteq \R$.

From Open Real Interval is Open Set, $\openint {-\dfrac 1 n} {\dfrac 1 n}$ is open in $\R$ for all $n \in \N$.

But:

$\displaystyle \bigcap_{n \mathop \in \N} \openint {-\dfrac 1 n} {\dfrac 1 n} = \set 0 = \closedint 0 0$

which is a closed interval of $\R$.

The result follows from Closed Real Interval is not Open Set.

$\blacksquare$


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