Topology is truly a twentieth century chapter in the history of mathematics. Although it saw its roots in work of Euler, Möbius, and others, the subject did not see its full flower until the seminal work of Poincaré and other twentieth-century mathematicians such as Lefschetz, Mac Lane, and Steenrod. Topology takes the idea of non-Euclidean geometry to a new plateau, and gives us thereby new power and insight.

Today topology (alongside differential geometry) is a significant tool in theoretical physics, it is one of the key ideas in developing a theoretical structure for data mining, and it plays a role in microchip design. Most importantly, it must be said that topology has permeated every field ofmathematics, and has thereby had a profound and lasting effect. Every mathematics student must learn topology. And physics, engineering and other students are now learning the subject as well.

Not only the content, but also the style and methodology, of topology have proved to be of seminal importance. Basic topology is certainly wellsuited for the axiomatic method. Hence the flowering of the schools of Hilbert and Bourbaki came forth hand-in-hand with the development of topology. The R. L. Mooremethod of mathematical teaching was fashioned in the context of topology, and the interaction was just perfect. Modern gauge theory and string theory, and much of cosmology, are best formulated in the language of topology.

The purpose of this book is to give a brief course in the essential ideas of point-set topology.