This book, intended to be covered in one semester by math majors, provides a rigorous introduction to the basic notions of topology. The text culminates with two “capstone” chapters, discussing two classical applications of abstract topology: (1) the classification theorem of compact and connected surfaces; (2) homotopy and fundamental groups.

The exposition is always clear and supported by a number of examples and illustrations. Moreover, the exercises proposed at the end of each section make the textbook suitable for a first undergraduate course on topology.

Topology is ubiquitous in mathematics and it would be impossible to write a book covering all its applications. The author’s choice is to discuss the classification of compact connected surfaces (up to homeomorphisms — that is, up to topological equivalence) and the fundamental groups of some typical topological spaces. On the other hand, topology provides also a powerful tool to deal with function spaces (this is the object of Functional Analysis); for this kind of applications we recommend the classical book by Fomin and Kolmogorov, [1].

After a short introduction on set theory, Paul Schick explains in a terse style, and with abundance of examples, the fundamental constructions in topology: products and quotients of topological spaces. As should be, metric spaces are introduced only at a later stage, as a very special instance of topological spaces (unfortunately, some other textbooks do not adopt the same strategy). Indeed, many topological spaces occurring in practice, e.g., in algebraic geometry, are not metric spaces. The reader familiar with calculus will notice that many notions, such as continuity, limit, convergence of subsequences, intermediate value theorem, are purely topological in nature, and can be therefore extended to more general situations. In fact, most of the fundamental theorems in analysis can be proved in the framework of Banach spaces (complete normed vector spaces). However, some definitions and theorems, such as uniform continuity, have all their significance only in metric spaces.

Two separate chapters are devoted to compactness and connectedness, probably the two most important topological properties of spaces (that is, properties invariant under topological equivalence). Again, it would be instructive to go a bit further and read about compactness in function spaces, the most remarkable result being Arzelà-Ascoli theorem (see [1]).

Chapter 8 deals with separation axioms. There is a hierarchy of separation axioms for topological spaces, the most familiar being the Hausdorff axiom. The real line is an example of Hausdorff space, that is, one can always ‘separate’ distinct points; however, quotients of Hausdorff spaces are often not Hausdorff. Moreover, the Zariski topology, i.e. the usual topology in algebraic geometry, is far from being Hausdorff. So it is important to be acquainted with such topologies, which can sometimes behave in a quite counterintuitive way. The reader will find several examples of non-Hausdorff topologies in the book.

The theory of homotopy, presented in the last chapter, has interesting connections with the theory of differential forms and with the Cauchy formula in complex analysis (see, for example, [2]). The concept of homotopy provides a rigorous meaning to the intuitive idea of ‘continuous deformation’ of paths in a topological spaces. The homotopy relation is actually an equivalence class on the set of continuous paths; it is a remarkable fact that these equivalence classes can be composed to get a group structure: the fundamental group.

As an example, the fundamental group of the surface pictured below is the free group with two generators:

This is a torus of genus 1 (i.e., it has one hole); according to the classification theorem, every compact connected orientable surface is homeomorphic to a torus of some genus g and the last chapter of the book describes the fundamental group of such surfaces.

Most of the material covered by the book originated from the groundbreaking work of Henri Poincaré and Leo J. Brouwer, at the beginning of the twentieth century. A number of historical remarks and footnotes enriches the text, giving the right chronological perspective to the theory.

**References:**

[1] Fomin, Kolmogorov, *Elements of the Theory of Functions and Functional Analysis* , Dover

[2] Rudin, Real and Complex Analysis , McGraw-Hill.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at [email protected].