Equivalence of definitions of profinite group

Statement
The following are equivalent for a topological group:


 * 1) It is the inverse limit of an inverse system of finite groups, each equipped with the discrete topology.
 * 2) It is a compact totally disconnected T0 topological group.

A topological group satisfying both equivalent conditions is termed a profinite group.

Facts used

 * 1) Hausdorffness is product-closed
 * 2) Hausdorffness is hereditary
 * 3) Tychonoff's theorem
 * 4) Closed subset of compact space implies compact
 * 5) Connectedness is continuous image-closed
 * 6) uses::Compact and totally disconnected implies every open neighborhood of identity contains an open normal subgroup
 * 7) uses::Compact implies every open subgroup has finite index
 * 8) Compact to Hausdorff implies closed

(1) implies (2)
Given: An inverse system $$G_i,i \in I$$ of finite groups with discrete topologies. $$G$$ is the inverse limit of the system.

To prove: $$G$$ is compact, $$T_0$$, and totally disconnected.

Proof

(2) implies (1)
Given: A compact $$T_0$$ totally disconnected group $$G$$.

To prove: $$G$$ is the inverse limit of an inverse system of finite groups.

Proof: