Definition talk:Subdivision (Real Analysis)/Rectangle

It is definitely a subrectangle of $P$, the subdivision. As an example, we have $R = \closedint 0 1 \times \closedint 0 1$ in $\R^2$. Let's use:
 * $P = (\set {0, 0.5, 1}, \set {0, 0.5, 1})$

We break the $x$- and $y$-axes $2$ pieces each ($\closedint 0 {0.5}$ and $\closedint {0.5} 1$). This gives us $2 \times 2 = 4$ subrectangles:
 * $\closedint 0 {0.5} \times \closedint 0 {0.5}$
 * $\closedint 0 {0.5} \times \closedint {0.5} 1$

and so on.

Being a subrectangle doesn't just mean being a rectangle and a subset of $R$. It means it is one of these finitely many blocks that the $P$ breaks $R$ into. --CircuitCraft (talk) 12:34, 19 August 2023 (UTC)
 * What does this sub mean? They are the rectangles belonging to $P$. Not sub-something of $P$. --Usagiop (talk) 13:02, 19 August 2023 (UTC)
 * I'm using the same terminology as my source, and I've seen similar terms used in other places as well. The sub is indeed stating that the rectangle is a piece of $R$, but we still regard them as "created by" the subdivision. --CircuitCraft (talk) 19:23, 19 August 2023 (UTC)
 * So, you say $\closedint 0 {0.5} \times \closedint 0 {0.5}$ is a subrectangle of the subdivision $P$ of $R$. OK, probably, this is really not wrong. --Usagiop (talk) 20:44, 19 August 2023 (UTC)