Suppose $(X_i, \varphi_{i,j})$ is a projective system, with $X_i's$ being compact Hausdorff spaces. Take $X=\varprojlim X_i$. Is it true that $X_i$ is "dense" in $X$, in that: $$X=\overline{\bigcup\limits_i X_i}$$ I'm fairly certain there's a similar result concerning inductive limits and their connecting maps, which is why I believe that it should hold. But it is not obvious that is the case.
Thanks!