Show that $\displaystyle X- E^{\circ} = \overline{X-E} $ where X is a metric space and E is a subset of X. Can we do this without using DMorgan's Laws (e.g. without using intersection of closed sets)?

Because you are taking away $\displaystyle E^{\circ} \subset E $ from $\displaystyle X $. Whereas $\displaystyle \overline{X-E} = X-E \cup (X-E)' $ where $\displaystyle (X-E)' $ is the set of limit points of $\displaystyle X-E $. So you are taking away more points from $\displaystyle X $. Adding back the limit points gives you $\displaystyle X- E^{\circ} $ (e.g. you are not adding any interior points).