Abstract: Mycielski related category and the continuum by proving that whenever there is a tower of comeager sets (that is comeager subsets of all finite dimensions), then there is a perfect set which threads through the tower (a distinct n-tuple pulled from the perfect set will land inside the corresponding comeager set). This has many applications in descriptive set theory. We can extend this work to study sizes beyond the continuum. In invariant descriptive set theory, the size of quotient spaces of the real line is studied through definable injections. In this setting the equivalence relation E_0 generates the next biggest size after the continuum. We will prove a variant of Mycielski's result for the quotient of the real line by E_0, and discuss applications and possible generalizations of this result.