Writing singleton as countable intersection/union of open sets I want to write $\{ x \}$ as a finite intersection of open sets. I came up with $\bigcap^{\infty}_{n=1} (x - \frac{1}{n}, x + \frac{1}{n})$, but I am not sure wether this is correct. If incorrect, is there another way to write a singleton as a countable union/intersection of open sets?

That is correct and indeed it expresses $\{x\}$ as a countable intersection of open sets.
On the other hand, there is no way of expressing it as a countable union of open sets, since each of those open sets would have to contain some interval around $x$, and therefore so would their union.
