DescriptionThe goal of this thesis is to discuss the Hausdorff Distance and prove that the metric space SX , which is the set of compact subsets of X = R n with the hausdorff distance is a complete metric space. In the first part, we discuss open r-neighborhoods and convexity. Proving some properties and providing examples for some definitions that are defined. This provides us with a background before we begin discussing the Hausdorff Distance. In the second part, we introduce the Hausdorff Distance and its properties. In conclusion, we go on to prove that the metric space SX is a complete metric space using everything that has been discussed previously in the thesis.