By J. van Mill

During this publication we examine functionality areas of low Borel complexity.

Techniques from common topology, infinite-dimensional topology, practical research and descriptive set theory

are essentially used for the research of those areas. the combo of

methods from a number of disciplines makes the subject

particularly attention-grabbing. between different issues, a whole and self-contained facts of the Dobrowolski-Marciszewski-Mogilski Theorem that every one functionality areas of low Borel complexity are topologically homeomorphic, is gifted.

In order to appreciate what's going, a fantastic historical past in

infinite-dimensional topology is required. And for reasonable quantity of data of size thought in addition to ANR thought is required. the required fabric was once partly coated in our past e-book `Infinite-dimensional topology, necessities and introduction'. a variety of what used to be performed there are available the following besides, yet thoroughly revised and at many areas accelerated with fresh effects. A `scenic' course has been selected in the direction of the

Dobrowolski-Marciszewski-Mogilski Theorem, linking the

results wanted for its evidence to attention-grabbing contemporary examine advancements in size thought and infinite-dimensional topology.

The first 5 chapters of this publication are meant as a textual content for

graduate classes in topology. For a direction in measurement thought, Chapters 2 and three and a part of bankruptcy 1 might be coated. For a direction in infinite-dimensional topology, Chapters 1, four and five. In bankruptcy 6, which bargains with functionality areas, fresh examine effects are mentioned. it will possibly even be used for a graduate direction in topology yet its taste is extra that of a study monograph than of a textbook; it's therefore

more compatible as a textual content for a examine seminar. The book

consequently has the nature of either textbook and a study monograph. In Chapters 1 via five, until stated

otherwise, all areas below dialogue are separable and

metrizable. In bankruptcy 6 effects for extra common periods of areas are offered.

In Appendix A for simple reference and a few uncomplicated proof which are vital within the publication were gathered. The e-book isn't really meant as a foundation for a direction in topology; its goal is to assemble wisdom approximately common topology.

The workouts within the booklet serve 3 reasons: 1) to check the reader's knowing of the fabric 2) to provide proofs of statements which are utilized in the textual content, yet aren't confirmed there

3) to supply more information no longer coated by means of the text.

Solutions to chose workouts were integrated in Appendix B.

These workouts are vital or difficult.