"General Topology I: Basic Concepts and Constructions Dimension Theory" is a comprehensive textbook edited by A. V. Arkhangel'skii and L. S. Pontryagin. Here is a detailed overview of the book:

Key Themes

1. Basic Concepts and Constructions:

   • The first part of the book outlines the fundamental concepts and constructions of general topology, including topics not previously covered in English language texts.
   • It covers various types of topological spaces such as linearly ordered spaces, metric spaces, metrizable spaces, and abstract praclosure operators.

2. Dimension Theory:

   • The second part of the book provides a survey of dimension theory, from its beginnings to recent developments.
   • It details the principal ideas and methods in dimension theory, including sketches of proofs for the main results.

3. Relationship Between Specialized Topics and Central Themes:

   • The authors emphasize the relationship between more specialized topics and the central themes of topology, giving the book a broad scholarly appeal that transcends narrow disciplinary lines.

Plot Summary

The book is structured into two main parts:

1. Part 1: Basic Concepts and Constructions:

   • This section introduces the foundational concepts of general topology, including definitions, properties, and examples of various types of topological spaces.
   • It covers topics such as separation axioms (T0, T1, T2), regular and normal spaces, and continuous maps of topological spaces.

2. Part 2: Dimension Theory:

   • This section provides a comprehensive overview of dimension theory, starting from its historical beginnings and moving to the most recent developments.
   • It includes detailed discussions on the principal ideas and methods in dimension theory, along with sketches of proofs for the main results.

Critical Reception

The book has been well-received for its comprehensive coverage of general topology and dimension theory. It is praised for being accessible to both undergraduate students and professional mathematicians, while also appealing to a broad scholarly audience.

Edition and Publication

The book was first published in 1990 by Springer-Verlag, and it is part of the Springer series on general topology.

Additional Information

• The book includes a translator's note and an author index, making it a valuable resource for scholars and students in mathematics and related fields.
• It features 15 figures to aid in the understanding of complex topological concepts.

In summary, "General Topology I" by A. V. Arkhangel'skii and L. S. Pontryagin is a seminal work in the field of general topology, providing a detailed and comprehensive introduction to both basic concepts and advanced topics in dimension theory. Its broad scholarly appeal and accessibility make it a valuable resource for a wide range of mathematicians and scientists.