The book exmaines closure spaces, an abstract mathematical theory, with special emphasis on results applicable to formal logic. The theory is developed, conceptually and methodologically, as part of topology. At the least, the book shows how techniques and results from topology can be usefully employed in the theory of deductive systems. At most, since it shows that much of logical theory can be represented within closure space theory, the abstract theory of derivability and consequence can be considered a branch of applied topology. One upshot of this appears to be that the concepts of logic need t be overtly linguistic r do logical systems need to have the syntax they are usually assumed to have. Audience: The book presupposes very little technical kwledge, but can probably be read most easily by someone with a background in symbolic logic or, even better, upper division or graduate mathematics. It should be of interest to logicians and, to a lesser degree, computer scientists and other mathematicians.