= Profinite group
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A <profinite group> is an <inverse limit> of finite discrete <groups>. For a directed compatible system, its elements are tuples $(g_i)$ whose coordinates agree under all transition <homomorphisms>. Multiplication and inversion operate coordinatewise. The <inverse-limit topology> is inherited from the product of the finite discrete spaces.
The product is <compact> and <Hausdorff>, and the compatibility equations define a closed <subset>. Conditions on finitely many coordinates give clopen neighbourhoods; in particular kernels of finite quotient projections form an identity-neighbourhood <basis>. Conversely, every compact Hausdorff totally disconnected <topological group> is recovered from its finite quotients by <open normal subgroups>.
For example, the additive <group> of <p-adic integers> is $\mathbb Z_p=\varprojlim_n\mathbb Z/p^n\mathbb Z$. Compatibility says that a residue modulo $p^{n+1}$ reduces to the preceding residue modulo $p^n$. The ordinary <integers> map densely into this <group>. Compactness belongs to the completed inverse limit; the <profinite topology on the integers is not compact>. <Fields and Galois Theory by James S. Milne>[chapter7, topological groups and inverse limits]{ref}
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