I think it is just 1; but I am also under the impression that it is just any open interval on $\mathbb{R}$.
Furthermore, I am trying to figure out how a compact interval $X = [0,1]$ inherite standard orientation from $\mathbb{R}$, from Guillemin and Pollack's Differential Topology.
To my understanding, "inherit from $\mathbb{R}$" means a linear transformation between $X$ and the basis of $\mathbb{R}$ whose determinant is positive, right?
Thanks.
In the context of topology, a basis is a collection of sets which, when closed under union, form the entire topology.
This is similar to the spirit of the linear algebra concept of "basis", in that you turn elements in the basis into all possible elements by applying the usual operation. However, it is quite difference in practice because a linear algebraic basis consists of elements in a vector space whereas a topological basis consists of subsets of a topological space (they are elements of something, but that something is not the space itself but the topology).
Therefore, a basis for reals is the set of all open intervals. By taking unions of intervals it is possible to form any open set; this is what is required.
EDIT: You mentioned that "an open interval" is a basis, this is not quite correct. Because in general on a topological space you have no notion of "translation", or "scaling", you will need* to include all of the open intervals in the basis. [[You could imagine that a topological vector space might benefit from a combined notion of "basis" that had these concepts packaged into it, but I don't know if this has standard terminology]].
Also, as Billy mentioned in the comments, a topological basis need not be unique.
Bonus: (*) Actually, you don't need quite all of them :) In fact, there is a countable basis for $\mathbb{R}$; this is one of the many things that makes the reals so awesome! I'll let you think about what that might be...