Veryify that the union of a co-dense set and a nowhere dense set is a co-dense set. Give an example to show that the union of two co-dense sets if not necessarily a co-dense set.
Note: A co-dense set $A$ in the topological $X$ denotes its complementary set is dense in $X$. And a nowhere dense set $A$ in $X$ if $\overline{A}$ is co-dense.
What I've tried: For the second question, I find an example. The real line $R$ is the whole space. $Q$ denotes all the rational numbers and $P=R-Q$ denotes all the irrational numbers. We see they are both co-dense sets. However, their union is $R$, and hence is not a co-dense. The first question is still tough for me.
Could anybody help me? New example is also welcome.