A First Course in Topology - Continuity and Dimension
This is a very nicely written elementary book on topology, which is (by the author's words) “suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study”. The book contains a lot of material for a one semester course. A leading concept (sometimes rather hidden) is that of dimension. To explain basic topics of general topology, the author uses the Peano curve (with a construction and a proof) and Brouwer's fixed point theorem in the plane with its proof based on the Sperner lemma (which is also proved). After introducing homotopy groups, the fundamental group of circle is computed and, as a corollary, Brouwer's fixed point theorem for disks is proved with its usual corollaries. The Jordan curve theorem is proved using the notion of an index and gratings. The remaining two chapters contain basics of simplicial homology theory with the proof of the generalisation of Brouwer's fixed point theorem and the Euler-Poincaré characteristic of complexes.