Must an uncountable set be somewhere dense? Have positive measure? The answers to these questions is perhaps surprisingly NO! With the infamous cantor set being the counter example. Throughout this post the term “Cantor set” would be used exclusively for the middle-thirds cantor set. The Cantor(middle thirds) set is constructed quite simply. Take the interval $[0,1]$, divide it into three equal line segments and cut the middle open interval out, then repeat the process with the remaining line segments, take each remaining line segment divide it into three equal line line segments, cut the middle open interval out and so...