The first two chapters are an overview of elementary set theory and beginning notions in topology, such as metric spaces, the product topology, etc. The authors assume that the reader has encountered these ideas before, and so they do not spend a lot of time explaining them. Calling one section a "potpourri of fundamental concepts" the authors define accumulation points, closure, etc. There are some interesting insights though that the authors give the reader, particularly in the discussion on homeomorphisms. They caution the reader, giving a pictorial example, that thinking of a homeomorphism as a stretching or deformation is a somewhat limited view. The example shows two objects that are homeomorphic but that cannot be deformed into each other in Euclidean 3-dimensional space.
The next two chapters cover connectedness, compactness, and metric spaces. The authors show what pathologies can arise in the consideration of connectedness, challenging the reader to find an example of a space with an explosion point. They also define the notion of a chain, a concept that proves to be very useful in geometric topology. The authors motivate its eventual application very well, their construction beginning with an arbitrary "entangled" collection of open sets, out of which the chain is systematically selected. The famous Knaster-Kuratowski example is discussed. For readers interested in moving on to dimension theory, this example is important, in that it is a one dimensional set that is not totally disconnected. Separation properties are discussed in Chapter 4, and again reflecting their prejudice for geometric topology, the authors define and discuss absolute retracts and absolute neighborhood retracts.
Things get very geometric in chapter 5, wherein topology of the Euclidean plane is discussed. The Jordan curve theorem is proved in detail, along with the Schoenflies theorem. The latter has to rank as one of the more amusing results in geometric topology, and its proof is a joy to construct. Then, in chapter 6, the authors return to the consideration of product spaces, and they also define and discuss inverse systems. An understanding of inverse systems is a must for readers intending to move on to algebraic topology. The dyadic solenoid, an important construction in the field of dynamical systems, is discussed geometrically and then shown to arise as an inverse limit.
Considerations of a more analytic nature appear in chapter 7, which deals with function spaces, weak topologies, and Hilbert spaces. The compact-open topology, important in many area of application, is discussed as a topology that guarantees that a sequence of continuous functions converges to a continuous limit. The weak topology is introduced as a generalization of the free union topology, and its importance in the study of cell complexes is pointed out.
The glueing and identification operations, so familiar from popular or more elementary expositions of topology, are discussed in chapter 8. These are the quotient spaces, and the authors discuss the cone and suspension of a space as examples. CW-complexes are then introduced and discussed in detail. This is followed in chapter 9 by a discussion of one of the most important of all topological spaces: continua. The Peano continua in the light of the Hahn-Mazurkiewicz are overviewed.
If the reader has studied differential geometry, then chapter 10 will be somewhat familiar, as it deals with paracompactness and partitions of unity, the later of which are used extensively to perform some very standard constructions in the theory of differentiable manifolds. Metrizability is also discussed, and the authors give an example of a Moore space that is not metrizable.
Chapter 11 gives an alternative view of convergence, wherein the authors discuss nets and filters. The pathologies that can arise for sequences in non-metric spaces are emphasized. Filters may be familiar for the reader who has studied mathematical logic, where they are used extensively.
Things heat up in chapter 12, wherein readers get to indulge in the intricacies of algebraic topology, a topic that has been hinted at in a few places in the first eleven chapters. Homotopy theory and the fundamental group make their appearance, as well as the notion of a direct limit. The higher homotopy groups are introduced in the problem sets. The reader versed in algebra will certainly appreciate this chapter, as well as the next one, which deals with covering spaces, which the authors mention is a topological analog of Galois coverings. Covering spaces allow the computation of the fundamental group, as well as being useful in many other applications.
Simplicial topology is introduced in chapter 14 as objects that have a local linear structure, and can thus be studied much more easily than more general types of spaces. Most readers will catch on very quickly to this category of spaces, due to its connection with notions from plane and solid geometry, and linear algebra. The simplicial approximation of maps is emphasized, with an elementary example of a continuous map that cannot be simplicially approximated given. A hint of the field of simple homotopy theory is given in the problem section, with the famous Bing's house with two rooms discussed.
The last 3 chapters of the book discuss applications of homotopy theory, a brief introduction to knot theory, wild sets, the classification theorem for 2-manifolds (which is proven in detail), and a brief introduction to n-dimensional manifolds. The authors discuss briefly the attempts to generalize the 2-D classification to 3-D, one being finding a proper generalization of the normal form, another being the removal of a maximal open 3-cell from the 3-manifold to obtain the "spine". The famous Poincare conjecture is related to these issues.
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For Jaclyn Smith's fans, this is truly a collectible item. The photos are great and you will not get disappointed. I've read other books written by Victoria Principal, Morgan Fairchild, Linda Evans, etc..but this book is truly the best among the best! Her beauty and exercise tips are on target and real. Plus the big factor is the style of writing--there's life, art and heart when she discussed things. It's not just selling you the goods but sharing you her passion and love of what she does.
Excellent !!!