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)