US not implies Hausdorff
This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., US-space (?)) need not satisfy the second topological space property (i.e., Hausdorff space (?))
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
Get more facts about US-space|Get more facts about Hausdorff space
Example of cofinite topology
Consider a countable set, say , equipped with the cofinite topology. With this topology, the set is a US-space, because by definition, the only convergent sequence are those that are eventually constant, with the unique limit being the eventual constant value. However, the space is not Hausdorff, because for any two distinct points , and open sets containing and , the open sets intersect. (another way of thinking of this is that the space is an irreducible space).