分形和维度理论中的泛空间

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.

历史上，对于度量空间而言，维度理论中寻找泛函的历史大约跨越了一百年的数学研究。历史可以自然地分为两个时期——经典的（可分的度量）和现代的（不一定可分的度量）。

经典理论现在已经通过几本书得到了很好的记录。这部著作是首个将1960年至2007年间的现代理论统一起来的书籍。如同经典理论，现代理论从根本上涉及单位区间。

特色包括 * 使用图形来阐明这些空间中的分形视角； * 深入覆盖了点集拓扑、映射理论、分形几何和代数拓扑等各个主题； * 最后一章包含综述，并为相关研究提供了历史背景，该研究还包括其他嵌入定理、图论以及闭合嵌入； * 每个章节都包含一个评论部分，提供历史背景并附有参考文献，作为连接文献的桥梁。

这部著作将对拓扑学家、从事分形几何的数学家以及数学史家有益。因为它专注于连通一般分形和维度理论中泛函之间的关系，这将成为研究生研讨会或自学的好教材——感兴趣的读者会发现许多相关的未解决问题，这些问题将进一步深入研究这些问题。

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