Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a vector space is the number of vectors in its basis. This to me makes sense, since according to my understanding, the dimension of some space or some set with some *'structure'* on it (Sorry, don't know how to put this) is the number of independent *'parameters'* needed to specify each *'point'* in it. Does this same understanding of dimension carry over to say a topological space or some other type of space? I read the following paragraph in **Basic Topology** by **M.A. Armstrong**

Taking the dimension of $X$ to be the least number of continuous parameters needed to specify each point of $X$ is no good. Peano's example shows that the square has dimension 1 under this definition.

This does my head in a little. What is the dimension of say a sphere or a torus? What does it mean to say that a surface is a two-dimensional, topological manifold? Or is it just the case that we have defined it to be that way, meaning that we say a set $A$ is said to have dimension $n$ if such and such is true? Is there some basic underlying principles guiding these definitons? The following definiton also confuses me if I think about it too much

Let $V$ be a vector space over an arbitrary ﬁeld F. The projective space $P(V)$ is the set of 1-dimensional vector subspaces of $V$ . If $dim(V ) = n + 1$, then the dimension of the projective space is $ n.$

In what sense does it have the dimension $n$? Why and how is it different from the orignal vector space? I'm cringing just thinking about it. Sorry, rambling now...