Dimension definition Consider the definition of the topological dimension of a topological space $X$ as the maximum of an increasing chain of closed irreducible subsets of $X$.
Is dimension a topological concept?
I mean here, If I define the dimension of a ring to be its Krull dimension, and then use this in schemes, am I still talking about the topological concept?
Now, if I have a manifold, and declare its dimension to be the dimension of the Euclidian spaces it resembles on a small scale, is this equivalent to its dimension as a topological space?
Hopefully, my questions make sense.

You're dealing with two notions of dimension which only coincide in a degenerate sort of situation. I would call the first notion the Krull dimension of a topological space (the supremum of lengths of chains of irreducible closed subsets), because, indeed, if $A$ is a ring, then its Krull dimension in the sense of commutative algebra coincides with the Krull dimension of $\mathrm{Spec}(A)$. 
Krull dimension is not so interesting for Hausdorff spaces like manifolds. Indeed, in a Hausdorff space, the only irreducible closed subsets are singletons, so all Hausdorff spaces have Krull dimension zero. Thus all manifolds have Krull dimension zero. In particular $\mathbf{R}^n$ has Krull dimension zero for all $n\geq 1$. But as a manifold its dimension is $n$. So the only time Krull dimension coincides with the usual notion of dimension for a manifold is if the manifold is zero-dimensional. 
