Typed topology and its application to data set

Abstract: The concept of $typed$ $topology$ is introduced. In a typed topological space, some open sets are assigned "types", and topological concepts such as closure, connectedness can be defined using types. A finite data set in $R^2$ is a typically typed topological space. Clusters calculated by the DBSCAN algorithm for data clustering can be well represent in a finite typed topological space. Other concepts such as tracks, port (starting points), type-p-connectedness, p-closure-connectedness, indexing, branches are also introduced for a finite typed topological space. Finally, $left-r$ and $up-left-r$ type open sets are introduced for data sets in $R^2$, so that tracks, port, branches can be calculated.