Regularity and Compactness Theorems

Given a smooth immersed hypersurface \(M\) in \(U\) (or just a subset of \(U\)), for any \(x \in \bar{M}\cap U\), we say \(x\) is a regular point of \(M\) if there exists a number \(r\gt{}0\) such that \(B^{n+1}_r(x)\subset U\) and \(\bar{M}\cap B^{n+1}_r(x)\) can be written as a union of finitely many smooth, compact, connected, embedded hypersurfaces \(\Sigma_i\) in \(B^{n+1}_r(x)\) such that \(\bar{\Sigma}_i\cap B^{n+1}_r(x)=\Sigma_i\). In general, we redefine \(M\) such that each point in \(M\) is a regular point and every regular point of \(M\) lies in \(M\).

The (interior) singular set of \(M\) is defined by