How does the cartesian product of Borel Sets work?

Question

I have that $$(\vec{a},\vec{b}] = (a_{1}, b_{1}] \times (a_{2}, b_{2}] \times (a_{3}, b_{3}] \times ... \times (a_{k}, b_{k}]$$ as a definition for Borel Sets. Now I know that for example in $R^2$. $$(a_{1}, b_{1}] \times (a_{2}, b_{2}] = \{(a_{1}, a_{2}], (a_{1}, b_{2}], (b_{1}, a_{2}], (b_{1}, b_{2}]\}$$ But how can a vector interval equal a set ? $$(\begin{bmatrix} a_{1} \\ a_{2}\end{bmatrix}, \begin{bmatrix} b_{1} \\ b_{2}\end{bmatrix}] = \{(a_{1}, a_{2}], (a_{1}, b_{2}], (b_{1}, a_{2}], (b_{1}, b_{2}]\}$$