Can there exists (non-trivial) manifolds of non-standard dimensions? Certainly, there do exist manifolds of dimension $n$ for any $n \in \mathbb{N}$ (as well as manifolds of countably many dimesnions). However, manifolds are not "first-orderizable" and hence we cannot use the compactness theorem here.
Hunch: I think that if $D$ is a non-principal ultrafilter, $I$ is countable, and if we let $\mathfrak{A_i} \models$ ZFC, then we will find manifolds of non-standard dimensions in $\prod_D \mathfrak{A}$.
Question: If this hunch is correct, how do we prove the existence of (non-trivial) manifolds in such a model? More concretely, if $n$ is a non-standard natural number, can we prove the existence of an $n$-sphere?
Edit: Question 2: Would $(S^1,S^2,S^3,...)\in \prod_D\mathfrak{A}$ be an $(1,2,3,4,...)$-sphere or a different (1,2,3,4,...)-manifold?