In $\mathbb{R}^n$ it is well-known that a smooth hypersurface $M$ (closed as a subset of $\mathbb{R}^n$) is the zero locus of a global smooth function (whose gradient is nonzero on $M$); from this one easily deduces that its complement has exactly two connected components.
Now let $X$ be a Banach spaces and let us say that $M\subset X$ is a closed hypersurface if
- $M$ is a closed, connected subset of $X$
- for each $x_0\in M$ we can find an open neighbourhood $V$ and a function $F\in C^1(V,\mathbb{R})$ s.t. $M\cap V=F^{-1}(0)$ and $DF(x_0)\in E^*\setminus\{0\}$.
It is still easy to show that $X\setminus M$ has at most two connected components. Are there examples of hypersurfaces with connected complement?
I suspect that if $X$ is a Hilbert space one could try to construct a global function again via line bundles, but the lack of compactness could be a problem..