Compact and totally disconnected implies every open neighborhood of identity contains an open normal subgroup

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Statement

Suppose G is a profinite group (i.e., it is a compact Hausdorff totally disconnected group) and U is an open subset of G containing the identity element of G. Then, U contains an open normal subgroup of G.

Facts used

  1. Compact neighborhood of identity in totally disconnected group contains compact open subgroup
  2. Intersection of conjugates by compact subset of open neighborhood of identity contains open neighborhood of identity

Proof

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