US not implies Hausdorff

From Topospaces
Jump to: navigation, search
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


It is possible to have a US-space (i.e., a topological space in which every convergent sequence has at most one limit) that is not a Hausdorff space.

Related facts


Example of cofinite topology

Consider a countable set, say \{ 1,2,3,\dots \}, 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 x,y, and open sets containing x and y, the open sets intersect. (another way of thinking of this is that the space is an irreducible space).