Nowhere dense notation confusion The text Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger defines nowhere dense as $X$ is nowhere dense in $M$ if $X^{-,-} = M$. What does this mean? 

In that book, $X^{-}$ is used to denote closure and $X'$ is used to denote complement. So $X^{-\prime}$ is the complement of the closure and $X^{-\prime -}$ is the closure of the complement of the closure (quite a mouthful).
Therefore this says that: A set $X$ is nowhere dense in $M$ if closure of the complement of the closure of $X$ is $M$.
See these two links.
