SOURCE:

Faculty: Physical Sciences
Department: Mathematics

CONTRIBUTORS:

Ilo, O. Alexander

ABSTRACT:

The major motivation for this study is the desire to fully appreciate the
landscape of weak topology, a key tool in functional analysis. Thus a constructive approach was taken which provided the key to open up the weak topology landscape and led to the exposure of several facts (erstwhile obscure) about weak topology and topology in general. It is proved that the
well known co-finite topology on any nonempty set is a tree of many cofinite-like topologies which were called semi-co-finite topologies. The concept of reducibility of topologies was introduced and it was proved (among other things) that the discrete topology of a set X cannot be reduced in a strong sense if the cardinality of X is greater than 2. We proved that all the factor spaces are discrete if a product topology is discrete in either finite or infinite dimensional situations. We established the conditions for the inducement by a weak topology on its range topological spaces (and inheritance by a weak topology, from its range spaces) of the properties of the lower separation axioms of T0, T1, and T2 (Hausdorff). We then obtained the conditions for the inducement by a weak topology on its range topological spaces (and inheritance by a weak topology, from its range spaces) of the properties of the higher separation axioms of Tychonoff, Normality, Regularity, Complete Regularity, and Complete Normality. We proved that any nontrivial weak topology is actually in the middle of a chain of pairwise strictly comparable weak topologies. We proved that every seminorm topology, which already is known to be locally convex, is actually the peak (maximum) of a sequence of pairwise strictly comparable non-locally-convex weak topologies which are generated by the given family of seminorms. We introduced the concepts of complement of a topology, complement topology and the supra of a topology.
We introduced and defined the concept of Exhaustive Topology and showed
that the supra of a topology cannot be discrete if the topology is not exhaustive. We also proved that no topology can exist between a topology τ and the supra τs of τ and have a distinct supra from τs. We defined discrete weak topology and indiscrete weak topology and showed that these weak topologies may not be trivial as topologies.