TITLE: Why study dimensions?
QUESTION [5 upvotes]: I am quite new to the forum so please feel free to correct/give pointers if I am posting something in the wrong place.
I have been perusing "Dimension Theory" by Witold Hurewicz & Henry Wallman and while doing so noticed I couldn't quite see the reason for 'studying/researching' dimension theory. As an undergraduate when I was taught the concept of dimension in terms of 'linear independence' and 'spanning' of vectors I never paid much heed to alternate/analytical formulations of dimension. Now having come across, say, the Hausdorff, Fractal, Assouad dimensions I am quite curious as to why mathematicians have focussed so much on this avenue. 
I would be very, very grateful for any pointers or links which can give me a little bit of 'general' motivation for studying dimensions i.e why study dimension theory. Specific examples from dynamical systems, metric geometry, number theory, set theory et cetera are very welcome too. 
Thank you, thanks a lot!
P.S the tags might be a little over the place - couldn't quite create new ones for metric geometry and assouad dimension.

REPLY [3 votes]: "Topological dimension" as defined by Hurewicz and Wallman is a topological invariant which lets you prove that many pairs of spaces are not homeomorphic to each other. 
For example, why are $\mathbb{R}$ and $\mathbb{R}^2$ not homeomorphic? 
The pedestrian answer is: $\mathbb{R}$ is separated by removal of any point, whereas $\mathbb{R}^2$ is not. 
But you might want to generalize: Why are $\mathbb{R}^m$ and $\mathbb{R}^n$ not homeomorphic when $m \ne n$?
To obtain a general answer, return to the special case and recast it in terms of dimension theory: 

$\mathbb{R}$ has dimension $1$ (because a point separates it, in fact every point locally separates, which is how dimension 1 is defined in some treatments; I believe the Hurewicz and Wallman definition is close to this but perhaps not exactly the same) 
but $\mathbb{R}^2$ does not have dimension $1$. 
and "dimension 1" is a topological invariant.

In general, topological dimension is an invariant assigned to topological spaces whose value is either a non-negative integer or $\infty$. Homeomorphic spaces must have equal topological dimension. Using it, the proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ when $n \ne m$ is that the dimension of $\mathbb{R}^n$ equals $n$ whereas the dimension of $\mathbb{R}^m$ equals $m$ (these must be verified, of course).
You've mentioned a few other types of dimension in your question, and each of them is an important invariant in a different context. For example, fractal dimension or "Hausdorff dimension" which I presume you mean, is a real-valued bi-Lipschitz invariant of metric spaces. Hausdorff dimension is what I would use to prove that the middle thirds Cantor set is not bi-Lipschitz equivalent to the every-other-fifths Cantor set, because their Hausdorff dimensions are $\log(2)/\log(3)$ and $\log(3)/\log(5)$ respectively, and these numbers are not equal.