How do I show that every collection of disjoint open sets in a separable space must be countable?
I am studying separable from this book and this was stated, and the prove was left to the reader. I am still trying to understand the material, so I have some problem with this, can anyone give me a light, please?
Best Answer
Since your space is separable there is a countable subset S which is dense. If your collection of open subsets is uncountable you could choose an element of S from each of these which would contradict the fact that S is countable.