Definition of a relatively open set:
$ D \subset K^N $ is a set. $U \subseteq D $ is relatively open in D if $$ U = \emptyset \quad$$ or $$ \forall x \in U \quad \exists \quad r > 0 \quad | \quad B(x,r) \cap D \subseteq U$$
What I want to know is: is there a set U with $x \in U \subseteq D $ such that $ B(x,r) \cap D \nsubseteq U$.
If $D = (0,2] $ and $ U =[1,2]$ and $x = 2$, then $B(2,r) \cap D = (2-r,2] \subseteq U$.
In the above example I dont see for any $x \in U$ where $ B(x,r) \cap D \subseteq U $ is not satisfied. Can someone please give examples of $D $ and $ U $ where $ B(x,r) \cap D \subseteq U $ is not satisfied.