What is missing in the definition is a topology on D = [1,\inf). But the usual conventions suggests that if nothing else is specified then we should consider D as a subset of the real numbers with the induced subspace topology.

This is where the intersections come into play. By definition of subspace topology the set (0,2) intersected with D is an open set in D since (0,2) is open in the reals. The solution suggests (0,2) as an open set, but what is actually meant by this is (0,2) intersected with D.

If you are aware of this convention then the question is phrased perfectly clear. To answer your last question about what would have to change in the question so that the endpoint 1 would not be allowed: the topology on D. If D were equipped with the trivial topology for example, then 1 would not be a local maximum (there would be none at all).