Hi, I was hoping someone might be able to help me understand the following statement:
"At a point p of a manifold M, a metric g has a unique signature, sig(g)=(p,n-p). Since, in terms of a chart, the components g_{ab} are continuous and since the matrix of components is invertible at each point, it follows that there is some open set O in M containing p such that sig(g)=(p,n-p) at each point in O."