The Branching Theorem and Semi-Cofinite Topologies

It is proved that every nonempty set X has a chain of topologies with the cofinite topology as its finest (maximum).For their semblance to, and yet differences from the cofinite topology, these other topologies in the chain are called semi-cofinite topologies.We proved that some of the semi-cofinite topologies in the chain are themselves the maxima of yet other sequences of pair-wise comparable semi-cofinite topologies on the nonempty set X.The cofinite topology lemma and the cofinite topology theorem are stated and proved.The entire exposition climaxed into what we finally called the Branching Theorem.The interesting meaning of the branching theorem is that every nonempty set is-topologically speaking-a tree of many branches and sub-branches of topologies that are pair-wise comparable.