In this note we discuss intertwined subsets of real line. We show that if two disjoint non-empty subsets and of real line possess the same boundary, then they are intertwined subsets if either and contain no intervals, or if they contain intervals, then they contain the end points of the intervals. In continue, by presenting new definition of intertwined sets of type two, we show that if and are intertwined sets, then either and are intertwined or they are intertwined sets of type two. Next we show that if is a function on with a unique infinite -limit set, then the -limit set is a cantor set. Finally, we discuss conditions that the set is dense in .