The definition of Y being connected in a topological space (X, tau) is that you can't find two non-empty, open and disjoint sets whose union is Y. This doesn't quite make much intuitive sense to me. For example, consider R with the usual topology. Then clearly, Y= [0,1] union [2,3] is not connected. That means you CAN find two non-empty, open and disjoint, sets whose union is Y. But what are they? I can't seem to think of 2 open sets in R whose union is [0,1] union [2,3].