So far we have been concerned with maps from an open subset of ℝ n into ℝ m . Soon we shall be considering maps from a set that is a subset of ℝ n into that very set, what are often called self map s of a set. For example, the map T:[0, 1]→[0, 1] given by Tx = 1 – x is a self map. A trivial example would be the identity map T given by Tx = x on any set X whatsoever. What we shall need is a property of a special kind of self maps called contractions or contraction maps of a closed subset of ℝ n (Theorem 4-1.6 below). Before proceeding to the theorem, we illustrate the ideas involved.