Completeness the hausdorff métric in compact space

This paper defines the Hausdorff metric on a closed and bounded subsets compact metric space. Throughout this is investigated by some topological properties as complete ness and total boundedness, for this two definitions is presented and also shown that they are equivalent finally shown that induced Hausdorff space is completeThis work is a reconstruction of part of the article Completeness and complete boundedness of the Hausdorf Metric Professor Jeff Henrikson. For this was a literature review to fill in the details on some of the demonstrations; It left open the possibility of inquiring about other properties induced Hausdorff space