This article is cited in 1 paper

Metrization of discrete topological spaces in the context of lattice theory. Part 1. On the normality of spaces

I. Yu. Torshin

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation

Abstract: A fundamental problem in machine learning and other modern methods of data analysis is the solution to the issue of generating metric distance functions (metrics) that would be adequate to the applied problems under study. The paper presents the results of a systematic analysis of the possibilities of metrization of discrete topological spaces using the concepts of lattice theory. A theorem on the regularity and normality of topological spaces arising in problems of recognition, classification, and numerical forecasting is proved. The regularity (according to Zhuravlev) of a set of precedents guarantees the normality of a topological space (separability axiom T4) and, consequently, the metrizability of this space. The author plans to put practical applications of the consequences of the theorem on regularity and normality presented in a separate paper that will make it possible to systematize the search for problem-oriented metrics which are most suitable for a particular applied problem.

Keywords: topological data analysis, lattice theory, algebraic approach of Yu. I. Zhuravlev and K. V. Rudakov, separation axioms.

Received: 08.12.2024
Accepted: 15.01.2025

DOI: 10.14357/19922264250111