The Significance of Understanding Product Topology in Data Science and Artificial Intelligence

• Author(s): Samuel Barasa ; Edward Videnyi
• Paper ID: 1704714
• Page: 753-758
• Published Date: 22-06-2023
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 6 Issue 12 June-2023

Abstract

This research investigates the properties and role of two different bases in defining the product topology on a Cartesian product of two topological spaces, X and Y. The first basis, denoted as ?, consists of all open sets of the form U × V, where U is open in X and V is open in Y. The second basis, denoted as ?, consists of all open sets of the form B × C, where B is open in X with respect to the product topology and C is open in Y with respect to the topology on Y. We establish that both bases are indeed bases for the product topology on X × Y and discuss their properties, including how they can be used to prove various results about the product topology. Our findings show that the basis ? provides an alternative way to define the product topology using open sets in X with respect to the product topology and open sets in Y with respect to the topology on Y, which may be useful in certain contexts.

Keywords

Product Topology, Data Science, Artificial Intelligence

Citations

IRE Journals:
Samuel Barasa , Edward Videnyi "The Significance of Understanding Product Topology in Data Science and Artificial Intelligence" Iconic Research And Engineering Journals Volume 6 Issue 12 2023 Page 753-758

IEEE:
Samuel Barasa , Edward Videnyi "The Significance of Understanding Product Topology in Data Science and Artificial Intelligence" Iconic Research And Engineering Journals, 6(12)