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

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

Let $\sequence {U_n}_{n \mathop \in \N}$ be an infinite sequence of open sets of $M$.

Then it is not necessarily the case that $\ds \bigcap_{n \mathop \in \N} U_n$ 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 $n \in \N_{> 0}$.

But:
 * $\ds \bigcap_{n \mathop \in \N_{> 0}} \openint {-\dfrac 1 n} {\dfrac 1 n} = \set 0 = \closedint 0 0$

which is a closed interval of $\R$.

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

Also see

 * Finite Intersection of Open Sets of Metric Space is Open