Profinite completion of the integers

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Definition

The profinite completion of the integers is defined in the following equivalent ways. Note that these definitions can be interpreted both as defining the group abstractly and as defining the group as a topological group:

1. The group obtained by taking the profinite completion of the group of integers (viewed as a discrete group).
2. The external direct product, over all prime numbers $p$, of the additive group of p-adic integers for the prime $p$. Note that we take the product topology from the topologies on these groups.

Arithmetic functions

Function	Value	Explanation
order of a profinite group	$\prod p^{\infty }$ for all primes $p$.	This is a supernatural number as is the convention for orders of profinite groups.

Group properties

Abstract group properties

Property	Satisfied?	Explanation
abelian group
aperiodic group

Topological group properties

Here, the topology is from the profinite group structure.

Property	Satisfied?	Explanation
profinite group	Yes	By definition
compact group	Yes	profinite groups are compact
T0 topological group	Yes	profinite groups are Hausdorff, hence $T_{0}$
totally disconnected group	Yes	profinite groups are totally disconnected
topologically cyclic group	Yes	$\mathbb {Z}$, the subgroup generated by the element that projects to 1 mod $p^{n}$ for all $n$, is a dense subgroup. In fact, this group can be thought of as a compactification of $\mathbb {Z}$.
topologically finitely generated group	Yes	Follows from being topologically cyclic.
connected topological group	No	It is nontrivial and totally disconnected.