Textbooks for topology dimension theory. I'm reading Munkres's  and notice the "dimension theory". I'm interested in it, yet I still have some questions:
(i) What's the motivation of (general topology) dimension theory?
(ii) Is dimension theory useful, i.e. has it solved some non-trivial problems?
(iii) Are there any good textbooks about this topic?
Thank you for your help!
 A: The primary motivation in the discovery of it (around 1910/1920) was trying to distinguish the different $\Bbb R^n$ topologically: it hadn't been shown in general that $\Bbb R^n \not\simeq \Bbb R^m$ when $n \neq m$ (some special cases were known), while there were examples of space filling curves, all sorts of fractal-like planar sets whose "dimension" was unclear etc. So some topologists defined functions "$\dim$" with integer values on topological (or metric, or even smaller classes) spaces that were topological invariants (so homeomorphic spaces had the same value ) hoping to find one for which they could show $\dim(\Bbb R^n) = n$. In the end several such functions were defined by different people (Lebesgue, Brouwer, Urysohn among them) and finally Brouwer succeeded (using deep theorems like his fixed point theorem and others) to show that indeed $\Bbb R^n$ had dimension $n$ for some dimension function. Soon people began generalising and unifying the different functions, especially in the realm of separable metrisable spaces, where most of such functions all coincide in value.
A good general overview (almost encyclopedic) is Engelking, theory of dimensions, finite and infinite. A good book on separable metric spaces only is Jan van Mill's  infite-dimensional topology, an introduction and prerequisites, who treats the basic dimension theory, Brouwer's fixed point theorem, and AR/ANR theory in just a few chapters, including a self-contained proof of the Jordan Curve Theorem. Too bad it's out of print nowadays.. (but a good library will have a copy).
The main problem it solves is the fact that the $\Bbb R^n$ are mutually non-homeomorphic but there are many other interesting results and still loads of open problems too. See Engelking's book for the frontier of research.
