A directed inverse limit of finite connected spaces is connected

Question

Apparently, a directed inverse limit of finite connected spaces is connected. Does anyone have a reference?

Answer 1

Suppose $X=\lim_i X_i$ is a directed limit of finite connected spaces.

Proof — As $X_i$ are finite connected spaces, by the Mittag-Leffler condition we can replace the directed system of $X_i$ by one in which all the transition maps are surjective. (Each $X_i$, being the image of a connected space, is still connected.) As a directed limit of nonempty finite sets is nonempty, the projections $p_i : X \to X_i$ are then surjective. If $U$ and $V$ are complementary nonempty clopens in $X$, they are quasi-compact since $X$ is. Therefore since the system is directed there is an $i$ so that $U = p_i^{-1}(A)$ and $V = p_i^{-1}(B)$ for opens $A,B$ in $X_i$. The opens $A$ and $B$ are disjoint since $p_i$ is surjective and $U$ and $V$ are disjoint. Also $A$ and $B$ partition $X_i$ for the same reason. QED