Fractals and Universal Spaces in Dimension Theory (Springer Monographs in Mathematics)

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9780387854939: Fractals and Universal Spaces in Dimension Theory (Springer Monographs in Mathematics)

Historically, for metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods - the classical (separable metric) and the modern (not-necessarily separable metric).

The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval.

Unique features include:
* The use of graphics to illustrate the fractal view of these spaces;
* Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology;
* A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings;
* Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature.

This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study - the interested reader will find many relevant open problems which will create further research into these topics.

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From the Back Cover:

For metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods ― the classical (separable metric) and the modern (not necessarily separable metric). While the classical theory is now well documented in several books, this is the first book to unify the modern theory (1960 – 2007). Like the classical theory, the modern theory fundamentally involves the unit interval.

 

By the 1970s, the author of this monograph generalized Cantor’s 1883 construction (identify adjacent-endpoints in Cantor’s set) of the unit interval, obtaining ― for any given weight ― a one-dimensional metric space that contains rationals and irrationals as counterparts to those in the unit interval.

 

Following the development of fractal geometry during the 1980s, these new spaces turned out to be the first examples of attractors of infinite iterated function systems ― “generalized fractals.” The use of graphics to illustrate the fractal view of these spaces is a unique feature of this monograph.  In addition, this book provides historical context for related research that includes imbedding theorems, graph theory, and closed imbeddings.

 

This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. It can also serve as a text for graduate seminars or self-study ― the interested reader will find many relevant open problems that will motivate further research into these topics.

Review:

From the reviews:

“The book is a research monograph reporting on an interesting area of research arising from the confluence of two streams: topology and self-similar fractals. It is written at a level that could be understood by graduate students and advanced undergraduates. It could be used for a seminar or introductory course for either topology or self-similar sets. ... The historical notes are informative and interesting. There is an extensive bibliography at the end of the book documenting the results and historical comments.” (J. E. Keesling, Mathematical Reviews, Issue 2011 b)

“The book under review is devoted to dimension theory in general. ... The book is completed by a useful appendix consisting of three parts, devoted to basics in topology, standard simplices in Hilbert spaces, and fractal geometry. So, it is accessible for all mathematicians, but should be of special interest for those working in topology or fractal geometry. The book contains a remarkable number of interesting historical remarks and colorful pictures.” (Uta Freiberg, Zentralblatt MATH, Vol. 1210, 2011)

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