It is often useful in the study and understanding of abstracts such as topological spaces to determine that one property does not follow from another.

In Counterexamples in Topology, Steen and Seebach, together with five students in an undergraduate research project at St. Olaf College, Minnesota in the summer of 1967, canvassed the field of topology for such counterexamples and compiled them in an attempt to simplify the literature.

For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set.

In her review of the first edition, Mary Ellen Rudin wrote: In his submission[2] to Mathematical Reviews C. Wayne Patty wrote: When the second edition appeared in 1978 its review in Advances in Mathematics treated topology as territory to be explored: Several of the naming conventions in this book differ from more accepted modern conventions, particularly with respect to the separation axioms.

This was a result of the different historical development of metrization theory and general topology; see History of the separation axioms for more.