Can there be an uncountable collection of open sets in $\Bbb R^n$?
My idea: Since every open set contains at least one rational number, I can match each open sets to rational numbers and rational numbers are countable. Thus collection of open sets must be countable.
My professor said "the union of an arbitrary collection of open subsets of $M$ is open, and arbitrary mean either finite, countably infinite or uncountably infinite". Here $M$ is a metric space.