Comparison Theorems for Weak Topologies (2)

Chika S. Moore, Alexander O. Ilo and Ifeanyi Omezi
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Chika S. Moore: Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State
Alexander O. Ilo: Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State
Ifeanyi Omezi: Department of Petroleum Engineering, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra

International Journal of Research and Innovation in Applied Science, 2025, vol. 10, issue 1, 227-235

Abstract: Weak topology on a nonempty set X is defined as the smallest or weakest topology on X with respect to which a given (fixed) family of functions on X is continuous. Let ð œ w be a weak topology generated on a nonempty set X by a family {fα: α ∈ ∆} of functions, together with a corresponding family of topological spaces. If for some , on is not the indiscrete topology and meets certain requirements, then there exists another topology on such that is strictly weaker than and is -continuous, for all . Here in Part 2 of our Comparison Theorems for Weak Topologies, We showed that not every weak topology ð œ w has a strictly weaker weak topology Ï„w1. We constructed important examples to show (a) that a weak topological system may not have a strictly weaker weak topology, (b) that a weak topological system can have a strictly weaker weak topology, and (c) that a weak topological system can have both comparable and non-comparable weak topologies. A further research agenda is (now) set to find out when and why we must use a particular weak topology (instead of the others) in any given context of analysis. Key Words: Topology, Weak Topology, Weak Topological System, Product Topological System, Chain of Topologies, Strictly Weaker Weak Topologies, Pairwise Strictly Comparable Weak Topologies

Date: 2025
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