Reducibility of Topologies

The concepts of weak or strong reduction of topologies are introduced. Closely related to these, and introduced as well, are the concepts of weak and strong base reduction of topologies. We also defined extensible topologies; and defined weak and strong base extension of topologies. We proved that there exists a topology γ, weaker than a weak topology τ, on X, which has a chain of strong reductions if one of the range spaces, say (Xα, τα) of τ, has a chain of strong reductions. It is proved that the usual topology of the set R of real numbers can be reduced in the weak sense to chains of infinite families of pairwise comparable topologies; and that the usual topology of R can neither be reduced in the normal sense nor in the strong sense. We proved that a weak topology has a chain of weaker topologies if one of its range topologies is reducible to a chain of topologies