Lattices of Reduction and Subset-Induced Topologies

Induced topologies have been studied only from the standpoint of a superset down to its subset to get what we call subspace topology. Here we turn the focus around and show that subsets can induce topologies on their supersets. Also, induced topologies have so far only been constructed by collecting the intersections of open sets of a superset with a subset. Here again we extend the focus and show that a superset will always induce topologies on their subsets through other means than by taking intersections of open sets with a subset. All these warrant further research into a more extensive and comprehensive study of induced topologies; to establish how some topological properties such as compactness, separation axioms, etc. are shared or inherited in the wider context of inducement of topologies. The concept of reducible topologies has been explored and published by the authors before [1]. Here we extend the research by proving that any pairwise comparable family F of subsets of a set X generates a reducible topology τ on X, and that the chain C of reductions of τ can be constructed in such a way that card(F) = card(C).