So I am learning topology on my own from a script. Earlier it defines what a co-finite topology is. Now I am trying to answer the question: "Consider R with the co-countable topology. Show that the closure of (0,1) is the whole space. On the other hand show there is no sequence in (0,1) converging to 2."
Now by analogy I would assume the co-countable topology is the collection of sets such that their complement is countable (and the empty set). However, I am very confused: since the complement of (0,1) in R is most certainly not countable, (0,1) should be closed. Therefore the closure of (0,1) would be (0,1), not R. However when I looked around on the internet a bit, it seemed like the closure is indeed R. What am I missing? Thank you in advance, I'm sorry for asking what I'm sure is a "dumb" question.