Topology is the branch of pure mathematics that studies the notion of shape. In the context of large, complex, and high dimensional data sets, topology takes on two main tasks, the measurement of shape and the representation of shape. One can measure shape related properties within the data, and create compressed representations of data sets retaining features which reflect the relationships among the points in the data set. The representation is in the form of a topological network or combinatorial graph. In the study of high dimensional and complex data sets, combinatorial representations provides a compressed representation of the data that retains information about the geometric relationships between data points. Also, the representations are a useful and simple way to examine the data, as well as understand the primary variables characterizing various subgroups. The three properties of topological analysis include: coordinate invariance, deformation invariance and compressed representations.
Topological Data Analysis (TDA) allows you to interact with and represent structured and unstructured data through a topological network. A topological network provides a map of all the points in the data set, so that nearby points are more similar than distant points and clarifies the structure of the data set without having to query it or to perform any algebraic analysis on only a subset of variables. In essence, one can discover the true meaning of the data by analyzing a compressed representation of the data set retaining all of the subtle features and data points that have a degree of similarity to each other.