Comparison Theorems for Weak Topologies (1)

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 τw be a weak topology generated on a nonempty set X by a family {fα,:α ∈ ∆} of functions, together with a corresponding family {(Xα, τα):α ∈ ∆} of topological spaces. If for some α0 ∈ ∆, τα0 on Xα0 is not the indiscrete topology and fα0 meets certain requirements, then there exists another topology τw1 on X such that τw1 is strictly weaker than τw and fα is τw1-continuous, for all α ∈ ∆. Here in Part 1 of our Comparison Theorems for Weak Topologies it is observed that: (a) the new topology τw1 on X deserves to be called a weak topology (with respect to the fixed family of functions) in its own right. Hence, we call τw1 a strictly weaker weak topology on X, than τw; (b) the usual weak, weak star, and product topologies have chains of pairwise strictly comparable (respectively) weak, weak star, and product topologies around them. All the necessary and sufficient conditions for the existence of τw1 in relation to τw are established. Ample examples are given to illustrate (at appropriate places) the various issues discussed.