Examples
The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology if and only if they are of the form U \ A where U is open in the Euclidean topology and A is countable. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff).
There are obscure examples of spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff. For details see Steen and Seebach.
Read more about this topic: Completely Hausdorff Space
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)