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.