If a tensor is found to be symmetric at an arbitrary point $X$; $B_{\mu\nu}(X)=B_{\nu\mu}(X)$. Can this be used to argue that $B_{\mu\nu}$ must be symmetric everywhere?
I'm following the derivation on Weinberg's Gravitation and cosmology section 13.4. After choosing an arbitrary point for which the Killing vector vanishes, a certain tensor was found to be symmetric then he argued ``Since X was arbitrary, this must hold everywhere''