relatively open sets 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
$$     \forall x \in U \quad \exists \quad r &gt; 0 \quad | \quad B(x,r) \cap D \subseteq U$$

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$. 

Example:

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.
Definition of a relatively open set, U must be in D, but D=(0,2] and U=[1,2] , U is not in D, this is D in U, your example is not a counterexample
Definition of a relatively open set, U must be in D, but D=(0,2] and U=[1,2] , U is not in D, this is D in U, your example is not a counterexample