Inverse limits of finite discrete groups are Hausdorff and totally disconnected, and compact assuming Choice

Every finite discrete group $G_i$ is Hausdorff because distinct points have disjoint singleton neighbourhoods. It is compact because an open cover has one member containing each point and [F3] makes those finitely many choices; the selected family is a finite subcover.

By [L2] and step 1.1, the full product is Hausdorff. Intersecting two disjoint ambient neighbourhoods with $L$ shows directly that the subspace $L$ is Hausdorff. Under the Axiom of Choice, [F1] and step 1.1 make the product compact; since $L$ is closed by [L1], [L3] makes $L$ compact.

By [L4], fixing finitely many coordinates gives a neighbourhood basis in $L$. Because every factor is discrete, each such cylinder trace is clopen. In particular, for $x\in L$ and an index $i$, the set $C_i(x):=\{z\in L:z_i=x_i\}$ is clopen.

If a connected subset $C\subseteq L$ contained distinct points $x\mathrm{\ne }y$, they would differ in some coordinate $i$. Then $C\cap C_i(x)$ and $C\setminus C_i(x)$ would be disjoint nonempty subsets that are both open and closed in $C$, a contradiction. Hence every nonempty connected subset of $L$ is a singleton, so $L$ is totally disconnected.