Introduction to Dimension Theory

Abstract

As with Chapter 1, the material presented in this chapter is rather advanced compared to the rest of this book. If you have never studied dimension theory before, you may find it difficult to understand the material in detail. I suggest that you browse through Chapter 8 without worrying about the details during the first reading; I hope that it will tell you something of what is significant in the theory. In Chapter 9 I have begun the subject again, with a self-contained and more elementary account. None of the actual results and definitions in Chapter 8 will be required for understanding the rest of the book.

Of all the theorems of analysis situs, the most important is that which we express by saying that space has three dimensions. It is this proposition that we are about to consider, and we shall put the question in these terms: When we say that space has three dimensions, what do we mean?

—Henri Poincare, quoted by Hurewicz and Wallman [1941]