Does Cartesian product refer to a segment or coordinate?

Question

Sorry for this rookie question, but we can not treat these ordered pairs as an interval e.g. treat $(a,b)$ as $\{x\in\mathbb{R}\mid a<x<b\}$, under any circumstances, right?

Answer 1

It seems you are confusing

• the ordered pair which is an element $(a,b)\in\mathbb{R}\times \mathbb{R}$

with

• the open interval which is a subset $(a,b)\subseteq \mathbb{R}$

beeing $a,b \in \mathbb{R}$, which have the same notation but are two completely different concepts.