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Geometric Aspects of General Topology (Paperback, Softcover reprint of the original 1st ed. 2013)
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Geometric Aspects of General Topology (Paperback, Softcover reprint of the original 1st ed. 2013)
Series: Springer Monographs in Mathematics
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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 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.
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