Definition of tangent space Terry Tao defines tangent space here as equivalence classes of continuously differentiable curves $\gamma : I \rightarrow G$ where $I$ is an open interval.
On the other hand, Wikipedia defines it as equivalence classes of curves $\gamma : (-1,1) \rightarrow M$.
I have 2 questions regarding this definition:
(i) Why does it have to be functions from open sets? Why can it not be paths $\gamma : [0,1] \rightarrow G$?
(ii) Are these definitions really equivalent?
The context in which I am asking this is Lie groups and Lie algebras which I just started to read about.
Many thanks for your help.
 A: (Let the usual symbols stand for the usual objects: $M$ a smooth manifold, $\gamma$ a smooth curve, etc.)
(1) One reflexively uses open sets because it's easiest to think of the tangent space at $x\in M$ as the set of velocities of curves passing through $x$.  For this, you need to be able to differentiate a curve at $x$, and to differentiate, you need an open neighborhood about the (a) preimage of $x$.
So you could certainly define the curves on a closed interval as long as the interval has interior and the (a) preimage of $x$ is in the interior of the interval.  It's just neater to define the tangent space with open intervals, because you're going to be working with the interior of the closed interval anyway.
(2) Yes.  I'm going to leave this as an exercise for you (commenter anon suggested the idea).  In fact, there are several other definitions for you to become familiar with: 
$T_xM$ is ...


*

*Equivalence classes of curves from an open interval (or the standard open interval) to $M$ passing through $x$;

*Equivalence classes of derivations of functions at $x$;

*Equivalence classes of pullbacks of tangent vectors in $\mathbb{R}^n$ via chart maps;

*The fiber over $x$ in the tangent bundle (which is then defined as the appropriate quotient of a disjoint union of $\mathbb{R}^n\times $(chart neighborhoods) ).


You should check that these are all equivalent to the Wikipedia/Tao definition.  This is just a long definition chase, so I don't feel compelled to outline details.
For more information, check out any introductory graduate differential topology textbook.  I can recommend Riemannian Geometry by Manfredo do Carmo, Semi-Riemannian Geometry with Applications to Relativity by Barrett O'Neill, and of course for a technically beautiful but less readable introduction there's always Introduction to Differentiable Manifolds and Lie Groups by Frank Warner.
PS- You mentioned you're thinking about this in the context of Lie groups.  Bonus points for seeing how $\mathfrak{g}$ is both $T_eG$ and the space of left-invariant vector fields.
(Edit: Added titles and full author names at request.)
