Clopen subsets of R?

Question

Show that the only closed and open subsets of R are the empty set and R itself.

Answer 1

Let S be a nontrivial subset of R that is both closed and open. Let T be its complement; T is also both closed and open.

Pick s in S and t in T. Suppose s
The intuition here is that a closed set is an open set plus its boundary, and if you have two open sets with no space between them than their boundaries intersect.