Removing some points from regular surface

Question

Is it possible to remove some point from a regular surface that the remaining surface be a regular value of some function? We know that inverse of regular point of function is regular surface but how about the inverse of this with removing some point from surface? I know the theorem state that every compact orientable surface is inverse image of regular value of some function. but I don't know that is it helpful to solve my problem.

Answer 1

If you're working in Hausdorff spaces, every level set of a continuous mapping is closed. If you remove a non-isolated point from a regular level set $S = \{x : f(x) = y_{0}\}$, leaving a non-closed set $S'$, then $S'$ is not a regular level set of a continuous function with the same domain.

You can remove an isolated point (e.g., if the target space is one-dimensional, so hypersurfaces are discrete), and you can remove any set $T$ you like by removing $T$ from the codomain of $f$ and removing the preimage $f^{-1}(T)$ from the domain of $f$, but these observations seem not to be in the spirit of the question.