If $S$ is a subset of $[0,1]\times[0,1]$ such that one point of the ordered pair is rational and the other is irrational or both are irrationals. Then which of the following is true?
a) $S$ is closed
b) $S$ is open
c) $S$ is connected
d) $S$ is totally disconnected
e) $S$ is compact
HINTS: It may be helpful to note that $S=[0,1]^2\setminus\Bbb Q^2$: $S$ is the set of points in the unit square that do not have both coordinates rational.
More specifically, answering (a) should give you the answer to (e) as well, and (a) and (b) should both be pretty easy to answer: does either $S$ or $[0,1]^2\setminus S$ contain a non-empty open ball? Answering either of (d) and (e) should give you the answer to the other. For (d) note that every horizontal line segment in $[0,1]^2$ whose $y$ coordinate is irrational lies completely in $S$, as does every vertical line segment whose $x$ coordinate is irrational.