I have consulted many books and notes on internet on the Riemann integral but in none of them have I found whether the upper bound of lower sums and the lower bound of higher sums depends on the type of intervals (open, closed or semi-open) choose to partition the integration interval $ [a,b] $.In particular let:
$$ s(f,\left\{I_k\right\}) = \sum_{k=1}^n \inf_{x\in I_k} f(x) m(I_k)$$
the lower sums and:
$$ S(f,\left\{J_l\right\}) = \sum_{l=1}^m \sup_{x\in J_l} f(x) m(J_l)$$
the upper sums. Generally these values depend on whether you choose each individual $I_k$ ($1\leq k \leq n$) and $J_l$ ($1\leq l \leq m)$ open, closed , open right or open left.
Intuitively upper bound of lower sums and the lower bound of higher sums do not depends on the type of intervals. If so, what is the proof? And if not, can you explain to me why? is the demonstration based on tagged partitions?