54. A set A is said to be nowhere dense in M if int (cI(A)) = Q. Show that {x } is nowhere dense if and only if x is not an isolated point of M. 55. Show that every finite subset of R is nowhere dense. Is every countable subset of R nowhere dense? Show that the Cantor set is nowhere dense in R. 56. If A and B are nowhere dense in M, show that A U B is nowhere dense. Give an example showing that an infinite union of nowhere dense sets need not be nowhere dense. 57. If A is closed, show that A is nowhere dense if and only if A is dense if and only if A has an empty interior. 58. Let (r, ) be an enumeration of Q. For each n, let I, be the open interval centered at rn of radius 2-", and let U = Un- In. Prove that U is a proper, open, dense subset of R and that U is nowhere dense in R