I was trying to understand the definition of a manifold. This question arised: is every manifold $M$ the inverse image of some $\Delta : \mathbb{R}^n \to \mathbb{R}$. The implicit function theorem, i understand, only says that yes, this indeed, holds, as long as the point of which the inverse image of $\Delta$ is calculated is regular.
Is the above statement true, or should state that "...every manifold is the inverse image of some family $\Delta_i...$"
One characteristic that I find in both the $S^1$ and $S^2$ (the only ones I can visualize) is that they can both be defined as the inverse image of a unique $\Delta$! What I want to understand is how a manifold would look if it were the inverse image of multiple functions $\Delta$. Would it have peculiar topological properties?
The answer is no. Note that such a (sub)manifold would have codimension 1 in $\mathbb{R}^n$. For example $\mathbb{RP}^2$ does not embed in $\mathbb{R}^3$ (for a rigourous proof see M.W.Hirsh, Differential Topology, chapter 4). But this counter example requires some tools so your question is not naive at all !
For another similar counter example, see here.