Definition:Subdivision (Real Analysis)/Rectangle

Definition

Let $R = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n}$ be a closed rectangle in $\R^n$.

Let:

$P = \tuple {P_1, \dotsc, P_n}$

where every $P_i$ is a finite subdivision of $\closedint {a_i} {b_i}$.

Then $P$ is a finite subdivision of the closed rectangle $R$.

Subrectangle

For every $1 \le i \le n$, let:

$P_i = \set {x_{i,0}, \dotsc, x_{i,m_i}}$

where:

$a_i = x_{i,0} < x_{i,1} < \dotso < x_{i,m_i} = b_i$

Then, for each $\sequence {k_1, \dotsc, k_n}$ such that:

$1 \le k_i \le m_i$

the closed rectangle:

$\closedint {x_{1,k_1 - 1}} {x_{1,k_1}} \times \dotso \times \closedint {x_{i,k_i - 1}} {x_{i,k_i}} \times \dotso \times \closedint {x_{n,k_n - 1}} {x_{n,k_n}}$

is a subrectangle of $P$.

Sources

• 1965: Michael Spivak: Calculus on Manifolds: $3$ Integration