I think you're misunderstanding the definitions at hand. Let

be given the subspace topology. Assume that

where

are open and disjoint. We may assume WLOG that

. Argue then that

otherwise

is a disconnnection of

. Then, argue that

since otherwise

is non-empty and open in

and since

and

then

and so

is a disconnection of

, thus

. Apply the exact same argument to show that

and thus

. Conclude that no disconnection of

exists.

*Remark* The same methodology (using induction) shows that if

is a sequence of connected subspaces of some topological space for which

then

is connnected.