Construct a set ScR such that S, the interior of S, the closure of S, the closure of the interior of S, and the interior of the closure of S are all distinct (ie no 2 of them are equal)
closure of S - smallest closed set containing S
interior of S - set of all points in S for which S is a neighbourhood
The Attempt at a Solution
I am really having trouble starting this...or more precisely, i'm having trouble seeing how this question is possible at all...How can a set, its interior, and its closure all be distinct? If a set is open, then its interior is simply equal to the set itself, so that leaves me with only closed sets to consider. But, if a set is closed, then it contains all of its boundary points, so the closure of S is equal to S...
so the only exception i could think of is the open/closed sets...
but so far the closest ive gotten to something that satisfies all of those things is the set of rational numbers
S = Q
int S = empty set
Closure of S = Real number line
but then closure of the interior of S = closure of the empty set = empty set
so thats equal to int S and doesn't work
and the interior of the closure of S = real number line = closure of S so that doesn't work either...help!