Comparison Theorems for Weak Topologies (2)

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