I am reading "Profinite Groups" by Ribes and Zalesskii and on page 9 it says that the projections of the nonempty inverse limit of a surjective inverse system are not necessarily surjective, but I can't come up with a counterexample. Here written out:
Given an inverse system $\{X_i,\varphi_{ij},I\}$ of topological spaces $X_i$ over poset $I$ with $X\not = \emptyset$ its inverse limit and the $\varphi_{rs}:X_r\to X_s$ surjective for all $s\le r$, this we call a surjective inverse system. Now the projections $\varphi_i:X\to X_i$ don't need to be surjective, they are if all $X_i$ are compact Hausdorff.
I am now looking for an example where the projections are not all surjective. I tried some stuff with taking $X_i = \mathbb R$ and playing with the maps between, but that didnt't work out. I think I found out the "reason" why it can go wrong, namely that we can take a $(x_i)\in X_i$ such that the inverse limit of $\{\varphi_{ji}^{-1}(x_i),\varphi_{jk}',J = \{j\in I |j\ge i\}\}$ ($\varphi_{jk}'$ the restriction) is empty, but I haven't been able to construct a counterexample.