Let $A_{1}, \ldots, A_{d}$ be a covering of $\mathbb{S}^{d}$. Then $A_{i} \cap\left(-A_{i}\right) \neq \emptyset$

Question

Below is an alternative formulation of Borsuk Ulam theorem

(v) Let $A_{1}, \ldots, A_{d}$ be a covering of $\mathbb{S}^{d}$ by closed sets $A_{i} .$ Then there exists i such that $A_{i} \cap\left(-A_{i}\right) \neq \emptyset$.

What does $(-A_i)$ means? i don't understand the meaning of the minus sign. Is it only saying take opposite of every element of $A$? i don't think so becvause then the statement seems false for $d=2$.

Answer 1

$-A = \{- x \mid x \in A \}$. It is the image of $A$ under the antipodal map $f(x) = -x$.