gn.general topology – Intersection of zero sets of continuous functions


Let the zero sets $F={x in mathbb{R}^n: f(x) = 0}$, $G = {x in mathbb{R}^n : g(x) = 0}$, where $f$ and $g$ are $m$-dimensional nonlinear continuous vector functions. Under some assumptions, these sets define hypersurfaces of zero measure in $mathbb{R}^n$. I was wondering:

  1. Are $F$ and $G$ always submanifolds embedded in $mathbb{R}^n$ or are there exceptions – in the latter case, are there conditions that guarantee that they are submanifolds ?
  2. What is the dimension of $Fcap G$ ? As pointed out here Measure of the intersection of two manifolds, if $F$ and $G$ are $(n-1)$-dimensional manifolds and their intersection is transversal, then $text{dim}(Fcap G) = n-2$. However, is there anything that can be said if the intersection is not transversal ? In general, I am interested in some sort of inequality $text{dim}(Fcap G) leq n-2$, assuming that $F subseteq G$, that $G subseteq F$ do not hold.