Lemma 2. The union of a collection of connected subspace of X that have a point in common is connected.
Lemma 3. The connected subsets of are intervals.
We already know that is connected with respect to K-topology on , which denoted as . By lemma 1 & 2, we only remain to show that (0,1) is connected with respect to . In standard topology on , (0,1) is connected by Lemma 3. Since is finer than the standard topology on , we need to check added open sets in (0,1) by . Since a basis element has the form , added open sets are simply unions of intervals. Now the whole (0,1) can be described as a union of connected intervals having intersections. Thus (0,1) is connected in by lemma 2 & 3.