Geometric Aspects of General Topology

This book is designed for graduate students to acquire knowledge of simplicial complexes, Dimension Theory, ANR Theory (Theory of Retracts), and related topics. These theories are connected with various fields in Geometric Topology, Algebraic Topology as well as General Topology. Except for the second half of the last chapter, this book is entirely self-contained. To make the ideas of proofs easier to understand, many proofs are illustrated with figures or diagrams. While exercises are not explicitly included, some results are provided with only sketches of proofs. Completing the proofs in detail is a good exercise for the reader. Researchers will also find this book very helpful, as it contains many important results not presented in usual textbooks, such as dim X × I = dim X + 1 for a metrizable space X; the difference between small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR property of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension-raising cell-like map; and a non-AR metric linear space. The last three subjects are linked to each other, demonstrating how deeply related the two theories are. Simplicial complexes are very useful in various fields of Topology and are indispensable for studying theories of dimension and ANR. Many textbooks deal with simplicial complexes, but none discuss in detail what is non-locally finite. For example, J.H.C. Whitehead's theorem on small subdivisions is very important, but its proof cannot be found in any other book. The homotopy type of simplicial complexes is discussed in textbooks on Algebraic Topology using CW complexes, but geometrical arguments using simplicial complexes are relatively easy. Many contents have been added to this edition to make it more comprehensive.