I'm trying to show an inverse map composed with its noninverse results in the identity in terms of the set map f:X-->Y between topological spaces when f is one to one function. If I define the inverse map of a set as the disjoint union of the inverse map of each point in the set in Y, then let that set be the image of a subset U in X. My problem is the change in indexing under the union operator. It looks that I have to go from "y is in f(U)" to "x is in U" when each inverse y is the singleton x (using one to one), but this appears to assume one to one correspondence between U and f(U), but isn't that what I'm trying to show?, so my thinking is running in circles. Maybe its technically illegal to index a union of points in a set by the points in the set. I'm sure there's a trivial solution to this, but I'm failing to see it.