# 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. Posted on Categories Articles