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

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

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

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

Proof
Consider the open real interval $\left({-\dfrac 1 n \,.\,.\, \dfrac 1 n}\right) \subseteq \R$.

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

But:
 * $\displaystyle \bigcap_{n \mathop \in \N} \left({-\frac 1 n \,.\,.\, \frac 1 n}\right) = \left\{{0}\right\} = \left[{0 \,.\,.\, 0}\right]$

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