This is a retitled, revised, refined, expanded edition of my topology book previously published in 1968 and 1988, and now available at the amazing price of $23.99. To order, please use the following link: http://www.booksurge.com/product.php3?bookID=GPUB05314-00001&affiliateID=A001356 Their Global Publishing System should allow more local printing and delivery. 538 pages. List of chapters: Some topology on the real line Topological spaces Connected spaces, compact spaces Identification spaces and cell complexes Projective and other spaces The fundamental groupoid Some combinatorial groupoid theory Cofibrations Computation of the fundamental groupoid Covering spaces, covering groupoids Orbit spaces, orbit groupoids Conclusion It combines a geometric and a categorical approach, where the latter allows clear analogies between areas of mathematics, such as topology and algebra. This is seen very clearly in the chapters on covering spaces and orbit spaces. Here are some examples of topics covered not available in other texts at this level: the initial topology on joins of spaces; the convenient category of k-spaces in the non Hausdorff case; a gluing theorem for homotopy equivalences; the Phragmen-Brouwer property, proved using the groupoid van Kampen theorem, and applied to the Jordan Curve theorem; the equivalence between covering maps of spaces and covering morphisms of groupoids; the fundamental groupoid of an orbit space by a discontinuous action of a group as the orbit groupoid of the fundamental groupoid. The book is graced by a cover by John Robinson and Ben Dickens. Ronnie Brown www.bangor.ac.uk/r.brown www.popmath.org.uk