This book is designed for graduate students to acquire knowledge of
dimension theory, ANR theory (theory of retracts), and related topics.
These two theories are connected with various fields in geometric
topology and in general topology as well. Hence, for students who wish
to research subjects in general and geometric topology, understanding
these theories will be valuable. Many proofs are illustrated by figures
or diagrams, making it easier to understand the ideas of those proofs.
Although exercises as such are not included, some results are given with
only a sketch of their proofs. Completing the proofs in detail provides
good exercise and training for graduate students and will be useful in
graduate classes or seminars.
Researchers should also find this book very helpful, because it contains
many subjects that are not presented in usual textbooks, e.g., dim X ×
I = dim X ] 1 for a metrizable space X; the difference between
the small and large inductive dimensions; a hereditarily
infinite-dimensional space; the ANR-ness 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 final chapter enables students to
understand how deeply related the two theories are.
Simplicial complexes are very useful in topology and are indispensable
for studying the theories of both dimension and ANRs. There are many
textbooks from which some knowledge of these subjects can be obtained,
but no textbook discusses non-locally finite simplicial complexes in
detail. So, when we encounter them, we have to refer to the original
papers. For instance, J.H.C. Whitehead's theorem on small subdivisions
is very important, but its proof cannot be found in any textbook. The
homotopy type of simplicial complexes is discussed in textbooks on
algebraic topology using CW complexes, but geometrical arguments using
simplicial complexes are rather easy.
