Comparison Theorems for Weak Topologies (3)

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. In an effort to construct and analyze strictly stronger weak topologies, only four weak topologies (namely, the Arens topology, the Mackey topology, the weakened topology, and the strong topology) have been compared before now. Also, all these weak topologies were constructed with an eye on only the polars of linear spaces (i.e. with a focus on only linear functionals on linear spaces). To that extent, the study of strictly stronger weak topologies before now can be described as a study of linear topological spaces. Here in Part 3 of our Comparison Theorems for Weak Topologies: We established and clarified the place of our comparison theorems in the context of the weakened topology, the Arens topology, the Mackey topology, and the strong topology that have up to now been compared. We showed that there are many other weak topologies between the four already compared weak topologies—constructible even by the use of polars. In particular, we showed that we can we find a weak topology stronger than the one hitherto known and referred to as the strong topology. Then we showed that if two weak topologies, generated by one family of functions on a set, are strictly comparable, then there exist in a range space two strictly comparable topologies which induce the weak topologies. The rather simplistic view that” The weak topology is Hausdorff” has been held by many authors for very long time. We changed that narrative here, as we stated and proved (with examples) that:” Some weak topologies are Hausdorff while others are not.”