Rockafellar Theorem 6.5 - Intersection of relative interiors

Question

I am currently reading through Rockafellar's "Convex Analysis" and I am trying to make sense of Theorem 6.5:

I understand most of the proof except for why the assumption of a finite index set is required to prove the second statement in the theorem. Rockafellar goes on to give the following example of why this is needed:

However I am struggling to understand why the two relative interiors are not the same in his counter example. I am concerned I am missing something fundamental regarding my understanding relative interiors.

I have left the other theorems used in the proof below.

Answer 1

Note that $\text{ri}[0,1+\alpha] = (0,1+\alpha)$ and intersection of $\text{ri}[0,1+\alpha]$ over all $\alpha>0$ would be $(0,1]$. However, note that $\text{ri}[0,1] = (0,1)$. Since, the later case does not include the point $1$, the equivalence fails to hold.