Why is $\mathbb R^n$ so important in algebraic topology? If you study the different tools from algebraic topology, you realise that most (if not all) of them are somehow meant to compare the space with Euclidean space:


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*Homotopy theory deals with maps from $S^n\subset\mathbb R^{n+1}$ into the space.

*Singular homology deals with maps from $\Delta^n\subset\mathbb R^{n+1}$ into the space.

*Simplicial homology only studies spaces built from simplices (which are subspaces of Euclidean space)

*Cellular homology does the same with $n$-cells.


All of these tools are excellent for studying spaces that are built up from Euclidean space using the operations from point set topology, identifying and crossing stuff here and there. But you would think that these spaces form an extremely small corner of the entire category of topological spaces. I would assume that there are many other (possibly not very “interesting,” from a concrete point of view) spaces that are very different from Euclidean ones and which are extremely hard to compare to them. So why does it seem that the entire toolbox from algebraic topology is only concerned with Euclidean-like spaces? I would assume that the explanation is some combination of the following, mainly the last one:


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*Euclidean spaces have some (to me unknown) universal property in the category of topological spaces that makes them very distinguished objects, a natural choice to compare others to; or

*Most of the concrete problems that we want to solve are naturally occurring in Euclidean space or something very similar.


Can one or both of these explain this issue?
 A: My algebraic topology professor had a slogan: "Point set topology looks at 'simple' properties of 'exotic' spaces whereas algebraic topology looks at 'exotic' properties of 'simple' spaces."
General topology lays out things like connected, path connected, locally connected, etc. Then you start to look at how these properties are related. You ask questions like: "If I'm sequentially compact, am I also compact? pseudocompact?" "What's a really weird space which has properties 4 and 6 but not 1 or 5."
Algebraic topology is trying to get at more fine tuned data. So we work in an "easy" setting: I'm connected, compact, ... heck, I'm a compact manifold. How can you tell me apart from this other nice space? I need some really finely tuned properties to do this. In comes homology / cohomology / homotopy etc.
When it comes down to it, the "nice" spaces like a torus or sphere are already "hard enough" to deal with, so we aren't as concerned with more difficult things.
Think of general topology like using a telescope and algebraic topology like using a microscope. You aren't going to begin by pointing your microscope up at the sky. Point it as your desk first.
Alternatively, we care a lot about $\mathbb{R}^n$ related spaces because that's where the lion's share of applications lie. :)
A: Some random answers:
1) Algebraic topology is still topology, and we care about things like continuity, smoothness, etc. that we wouldn't get if we were working in the more general setup of, say, algebraic geometry. 
2) Algebraic topology is a technical area of mathematics, and relaxing $\mathbb{R}^n$ to a more general space will cause some important things to break. For example, one of the reasons CW-complexes are popular (aside from a combination of generality and simplicity) is that Whitehead's theorem holds for them; that's not true for a general category.
3) The $S^n$ in homotopy theory isn't arbitrary; it comes from the fact that $S^{n+1}$ is the suspension of $S^n$. In a sense, $S^0$ is the "simplest" nontrivial pointed space; it's just a basepoint with one more point added.
4) We still want to study manifolds even in algebraic topology, and manifolds are built out of copies of $\mathbb{R}^n$. (Looking at manifolds based on, say, general Hilbert spaces works well but does differ from the finite-dimensional case.)
5) We don't restrict ourselves to topological $n$-manifolds. There's a lot of work in algebraic topology about finding conditions ensuring (or preventing)  that a "reasonably nice" space is actually a topological manifold. 
