Typed Topological Structures Of Datasets

• URL: http://arxiv.org/abs/2508.14008v1
• Date: Tue, 19 Aug 2025 17:14:13 GMT
• Title: Typed Topological Structures Of Datasets
• Authors: Wanjun Hu,
• Abstract summary: A datatset $X$ on $R2$ is a finite topological space.<n>In this article, we develop a special set of types and its related typed topology on a dataset $X$.<n>Such structures provide a platform for new algorithms for problems such as calculating convex hull, holes, clustering and anomaly detection.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \cite{carlsson}. In \cite{hu}, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a dataset. It is a new method from the general topology perspective. A typed topological space is a topological space whose open sets are assigned types. Topological concepts and methods can be redefined using open sets of certain types. In this article, we develop a special set of types and its related typed topology on a dataset $X$. Using it, we can investigate the inner structure of $X$. In particular, $R^2$ has a natural quotient space, in which $X$ is organized into tracks, and each track is split into components. Those components are in a order. Further, they can be represented by an integer sequence. Components crossing tracks form branches, and the relationship can be well represented by a type of pseudotree (called typed-II pseudotree). Such structures provide a platform for new algorithms for problems such as calculating convex hull, holes, clustering and anomaly detection.

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