1- show that the empty set is connected
2- is the set of Q ( the rationals , with the usual topology R ) connected ??? >>> prove it !!!
Do you know the definition of connected set?
Apply it to the empty set.
Consider the sets $\displaystyle H = \mathbb{Q} \cap ( - \infty ,\sqrt 2 )\,\& \,K = \mathbb{Q} \cap (\sqrt 2 ,\infty )$.
Are they nonempty separated sets the union of which is the rationals?