Introduction To Topology Mendelson Solutions [portable] -

• Assume ( (0,1) = A \cup B ) disjoint nonempty open in subspace topology.
  Pick ( a\in A, b\in B ) with ( a<b ). Let ( c = \sup x\in [a,b] : [a,x] \subset A ).
  Show ( c ) cannot be in ( A ) or ( B ) without contradiction.
  Thus no separation exists.

While "solution manuals" are often sought for quick answers, the true essay-worthy point is the process of derivation Introduction To Topology Mendelson Solutions

Chapter 4: Connectedness – Explores one of the two most critical topological properties, including applications to the real line. Master Topology with Bert Mendelson: A Guide to