Open Rectangles Closed under Intersection

Theorem
Let $\left(({\mathbf a \,.\,.\, \mathbf b}\right))$ and $\left(({\mathbf c \,.\,.\, \mathbf d}\right))$ be open $n$-rectangles.

Then $\left(({\mathbf a \,.\,.\, \mathbf b}\right)) \cap \left(({\mathbf c \,.\,.\, \mathbf d}\right))$ is also an open $n$-rectangle.

Proof
From Cartesian Product of Intersections: General Case, we have:


 * $\displaystyle \left(({\mathbf a \,.\,.\, \mathbf b}\right)) \cap \left(({\mathbf c \,.\,.\, \mathbf d}\right)) = \prod_{i \mathop = 1}^n \left({a_i \,.\,.\, b_i}\right) \cap \left({c_i \,.\,.\, d_i}\right)$

Therefore, it suffices to show that the intersection of two open intervals is again an open interval.

Now let $x \in \left({a_i \,.\,.\, b_i}\right) \cap \left({c_i \,.\,.\, d_i}\right)$.

Then $x$ is subject to:


 * $x > a_i$ and $x > c_i$, i.e., $x > \max \left\{{a_i, c_i}\right\}$
 * $x < b_i$ and $x < d_i$, i.e., $x < \min \left\{{b_i, d_i}\right\}$

and we see that these conditions are satisfied precisely when:


 * $x \in \left({\max \left\{{a_i, c_i}\right\} \,.\,.\, \min \left\{{b_i, d_i}\right\}}\right)$

Thus, we conclude:


 * $\left({a_i \,.\,.\, b_i}\right) \cap \left({c_i \,.\,.\, d_i}\right) = \left({\max \left\{{a_i, c_i}\right\} \,.\,.\, \min \left\{{b_i, d_i}\right\}}\right)$

showing that indeed the intersection is an open interval.

Combining this with the above reasoning, it follows that indeed:


 * $\left(({\mathbf a \,.\,.\, \mathbf b}\right)) \cap \left(({\mathbf c \,.\,.\, \mathbf d}\right))$

is an open $n$-rectangle.