Together with their graduate students, Steen and Seebach canvassed the field of topology for a wide grouping of topological counterexamples. If you're wondering whether one property of topological spaces follows from another, this book can usually provide a counterexample if it's false. For example, is there an example of a first-countable space which is not second-countable? The answer is yes, as counterexample #3 (the discrete topology on an uncountable set) is the first to show.
Steen and Seebach were working on the metrization problem, which asks which topological spaces can be made into metric spaces. This problem had inspired topologists to define a wealth of topological properties, some of which metric spaces had and some of which they did not. By comparing and contrasting these properties in a single refernce, Steen and Seebach simplified the relevant literature. Several other "Counterexamples in ..." books and papers have followed.
Note that several of the naming conventions in this book differ from more modern conventions (including those in Wikipedia), particularly with respect to the separation axioms. Steen and Seebach exchange the meanings of T3, T4, and T5 with those of regular, normal, and completely normal. They also exchange the meanings of completely Hausdorff with Urysohn. This was a result of the different historical development of metrization theory and general topology; see History of the separation axioms for more.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants.
Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing).
In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
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