I am trying to produce an example of surjective inverse system of $C^*$ algebras with empty inverse limit in analogy to Waterhouse example. So far I was trying something weaker namely to find surjective inverse system for which at least one projection is not surjective, I tried it like that:
take no-normal, locally compact, Hausdorff space $X$ and consider the inverse system of $C(K)$'s where $K$ are compact subspaces of $X$ and maps are projections. My guess (I can prove it in case X can be exhausted by compact sets) is that $$\underline{lim}_{K \subset X}C(K)=C^b(X)$$ and maby for concrete example of $X$ some function on some compact subset can't be extended to $X$ (I am not sure we can assume it will happen for every no-normal space if we considere only compact sets). Any suggestions if that idea is leading to solution (X can be found for which it will work) or new path are welcomed.