TITLE: Is a topology determined by its convergent sequences?
QUESTION [24 upvotes]: Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences?

REPLY [3 votes]: I like the following example. Let $D$ be an infinite discrete space and $\beta D$ its Stone-Čech compactification. Then a sequence in $\beta D$ converges if and only if it is eventually constant. Thus convergent sequences do not distinguish between the compact topology of $\beta D$ and the discrete topology on its underlying set. Obviously - the same holds for every refinement of $\beta D$.