On page 16 of The Large Scale Structure of Space-Time (1973) by Hawking and Ellis, they describe the basics of tangent spaces. This line appears near the top of the page:
Thus the tangent vectors at $p$ form a vector space over $R^1$ spanned by the coordinate derivatives (∂/∂xj)|p...
Shouldn't it be $R^n$ instead of $R^1$?
Is it simply a typo?
No, not a typo. The entire phrase “$V$ is a vector space over a field $\Bbb{F}$” should be thought of as a whole; the precise definition is as a tuple of information $(V,+,\cdot,\Bbb{F})$, where $+:V\times V\to V$ and $\cdot:\Bbb{F}\times V\to V$ are given functions (called the “operations” of vector addition and scalar multiplication respectively) such that a list of 8 or so axioms are satisfied. See What exactly is a "vector" in math (in terms of vector spaces)? to get over the hurdle from the usual introductory definition of vectors (as pointy arrows with length and direction), into the more abstract and general treatment (take a look in page 1 of any proper linear algebra textbook, e.g Friedberg, Insel, Spence, for a more detailed description).
Next, completely separate from the definition of a vector space over a field, is the definition of dimension of a vector space (over a given field), $\dim_{\Bbb{F}}V$.
A-priori separate from this notion of dimension of vector space, is the notion of dimension of a smooth manifold.
What that little portion of Hawking and Ellis’ text is telling you is that if you consider a smooth $n$-dimensional manifold $M$ (the word “dimension” here being used in the sense of manifolds), and a point $p\in M$, then the set of tangent vectors, $T_pM$, actually forms a vector space over $\Bbb{R}$, and furthermore if you consider a coordinate chart $(U,x)$ around the point $p$, then the set of tangent vectors $\left\{\frac{\partial}{\partial x^i}(p)\right\}_{i=1}^n\subset T_pM$ forms a basis for $T_pM$. Therefore, $T_pM$, as a vector space has dimension $n$, which is also the same as the manifold’s dimension.
