Example 2.2 from Boyd and Vandenberghe's Convex Optimization:
The textbook says that the interior of the unit square
$$\{x \in \mathbb{R}^3 | -1 \leq x_1 \leq +1, -1 \leq x_2 \leq +1, x_3 = 0\}$$ is empty.
The interior of a set is defined as the union of all open subsets of the set. It seems to me that in this case, the interior should be the same as the relative interior noted in the example (the unit square, excluding the boundary). Can someone explain why this is not the case?
Best Answer
Because every open ball centered at a point of the unit square contains points such that their third coordinate is not $0$. Of course, no such point belongs to the unit square.