Help understanding manifolds and topological spaces I'm used to think about linear algebra with matrices and vectors, I don't have particular problems with geometry either, I'm having hard times understanding what is the meaning of a manifold and a topological space in the math world.
I need this because a literally can't read algorithms about manifolds and topology and I realize that the meaning of this 2 words are really different from what you get used to think in the 3D world (digital sculpting, 3D editors and similar).
I also realized that this topics are often more abstract and more analitical than the geometry itself that they are trying to analize and process, this algorithms about topology are often about something that comes first than the mesh itself, so I think that the linear algebra it's not gonna help me here and I can't understand why a manifold is so different from a topological space; can you name a topological space that is not a manifold ? What is space that is not an Euclidean space ? What kind of topics I should study to understand this ?
The only thing that I realize is that Algebraic topology is more abstract than Differential topology, and in the end ( I'm interested in algorithms about 3D meshes for now, so "manifolds only" ) I probably need differential topology, but I'm not that sure about how to start, and I'm just writing this to let you know in what kind of confusion I am, I don't want to deverge from the real question.
I would appreciate a suggestion for where to start studying this having algorithms about 3D mesh elaboration as a final target, thanks.
 A: A manifold of dimension $n$, or $n$-manifold, is something that looks like $\mathbb{R}^n$ up close, but might curve back on itself. E.g. a sphere is a 2-manifold, an inner tube is a 2-manifold, and the space we live in is (perhaps) a 3-manifold.
Notions of "up-closeness" and "looking like" are treated mathematically by topology. A (general) topological "space" is something in which it makes sense to talk about "up-closeness." For instance, a circle is a topological "space," and so is an inner tube. These are both manifolds; a circle is a 1-manifold, and a inner tube is a 2-manifold.
For an example of a topological space that is not a manifold, consider the letter "A". You cannot make it look like a single line / or \ or - up close to either of the two junctions /- or -\ . So the letter "A" is not a manifold.
The first topological spaces that were studied as such were polyhedra. A polyhedron is, intuitively, composed of some polygons glued together along their boundaries. From what I can tell by a cursory glance on Wikipedia, 3D meshes are polyhedra lying inside $\mathbb{R}^3$.
The topology of polyhedra is called "piecewise-linear" topology, or "combinatorial" topology. PL topology is different from differential topology, which is what is treated in most texts on manifolds.
So I would recommend reading any of


*

*Classical Topology and Combinatorial Group Theory by John Stillwell,

*Geometric Topology in Dimensions 2 and 3 by Edwin Moise,

*A Textbook in Topology by Seifert and Threlfall,


if you want an introduction to PL topology. Also, read Proofs and Refutations by Imre Lakatos if the motivation for the definitions is unclear.
Of course, if you're intending to approximate a smooth surface in $\mathbb{R}^3$ with a 3D mesh, then the smooth surface itself will be a differentiable 2-manifold, and you might want to learn differential geometry. The most popular text on this subject is do Carmo's book Differential Geometry of Curves and Surfaces. I find it messy, and I much prefer Morita's book Geometry of Differential Forms; however, this is more advanced, and doesn't discuss curvature until late in the book. O'Neill's Elementary Differential Geometry strikes a balance between them, perhaps.
In any case, any such text will cover enough differential topology to make geometry possible---in particular, such a text will define "manifold", "tangent space", "vector field", "Riemannian metric", and so on.
That should get you properly introduced.
P.S. Here's something to think about: is the following a polyhedron? Is it a 3D mesh? Why or why not?

A: I recommend Loring W. Tu's An Introduction to Manifolds (Universitext).   And I've reviewed several other texts on the subject and found this one to be the best.
