Let $A\subset X$. $int(\bar{A})\neq\phi$ iff $A$ is dense in some open subset of $X$.
Using this I am unable to produce an open set in $\mathbb{R}$ with the usual metric for which $A=\{8\}\bigcup(0,1)$ is sitting inside as a dense subset of this open set. Can anyone please help me to get out of this block or rectify my understanding?.
What the statement says is: $int (\overset - {A})$ is non-emty if and only if there exists a non-empty open set $U$ in $X$ such that $A \cap U$ is dense in $U$. In your example you can take $U=(0,1)$.