1.Introduction Clustering is a process of sorting objects,elements or data into groups according to their similarity or dissimilarity.In this thesis,topological foundation and several approaches are going to be explained.2.Definition In a set of data,a cluster is a group of elements in which the elements are more similar to each other than elements in other clusters.We can put these elements into a metric space to measure the similarity between them by a"distance".This function's purpose would be measure the similarity between two elements.