Abstract

Compactness is considered one of the most important properties to be possessed by a topological space X. Yet it is also believed that it is not easy to come across a compact space. So if a space X is not compact, mathematicians would look to see if there is any way through which what approximates to compactness can be achieved for X. This article breaks the invincibility of compactness; it establishes that any nonempty set X has as many topologies as (at least) the cardinality of X, each of which makes X compact. Furthermore, we showed that if Card(X) = n, then there exists a chain. C = {τ_α : α ∈ ∆} of pairwise comparable topologies on X such that (X, τ_α) is a compact topological space, for each α ∈ ∆, and Card(∆) = Card(C) ≥ n. In short, we bumped into a Proliferation of Compact Topologies for any nonempty set X. In fact, if A is a subset of a nonempty set X, then there exist as many topologies as the cardinality of X namely {t_x: x∈X}, on X, with respect to each of which the subset A is compact. This is a crucially strange result.

Keywords: Topology, Compact topology, Finer, Coarser, Weaker and stronger topologies, Cofinite topology, Semi-cofinite topologies, Comparable topologies, Chain of topologies