Stone-Čech compactification of the natural numbers ($\beta \mathbb{N}$)

The Stone-Cech compactification $\beta \mathbb{N}$ can be identified with the space of ultrafilters on $\mathbb{N}$. Its basic clopen sets are $\bar{A}=\{\mathcal{U}:A\in \mathcal{U}\}$.

This is the natural-number-space case of Stone-Čech compactification.

Every convergent sequence in the Stone-Čech compactification of the natural numbers is eventually constant. If distinct ultrafilters $U_n$ converged, isolate each from the other sequence terms using convergence and finitely many separating clopen sets. Finite disjointification gives disjoint $B_n\in U_n$. The union of the even-indexed $B_n$ then belongs exactly to the even-indexed ultrafilters, contradicting convergence of membership in a clopen set.

For $A\subseteq \mathbb{N}$, the basic set $\hat{A}=\{U:A\in U\}$ in the Stone-Čech compactification of the natural numbers is a clopen set, since its complement is $\widehat{\mathbb{N}\setminus A}$. Distinct ultrafilters are separated by complementary basic sets. A basic cover with no finite subcover would give complements with the finite intersection property, whose generated filter on a set extends to an ultrafilter missing the entire cover. Thus the space is a compact Hausdorff space.

For ultrafilters $\mathcal{U},\mathcal{V}\in \beta \mathbb{N}$, define $\mathcal{U}+\mathcal{V}$ byThis operation is associative, and each right translation $\mathcal{U}\mapsto \mathcal{U}+\mathcal{V}$ is continuous, so $\beta \mathbb{N}$ is a compact Hausdorff left-topological semigroup.

An additive idempotent ultrafilter satisfies $\mathcal{U}+\mathcal{U}=\mathcal{U}$. On positive integers it is necessarily nonprincipal, since a principal ultrafilter at $n$ adds to itself to give the principal ultrafilter at $2n$. The idempotent-ultrafilter star-set lemma converts this algebraic property into a recursive construction of finite sums. If zero is included, the trivial principal idempotent at zero must be excluded for that increasing-sequence application.

For a sequence of positive integers, the nested closures of its tail finite-sums sets have nonempty compact intersection in the Stone-Čech compactification of the natural numbers. This intersection is a subsemigroup. For a sum $s$ from one tail, all sums supported beyond a chosen finite representation of $s$ lie in that tail translated by $-s$. The ultrafilter addition formula then establishes closure under addition. The Ellis–Numakura lemma supplies an idempotent ultrafilter containing every tail finite-sums set and hence any set containing the full finite-sums set.

An ultrafilter on the positive integers selects exactly one residue class modulo a fixed positive integer $q$. Under addition on the Stone-Čech compactification of the natural numbers, selected residues add. An idempotent ultrafilter must therefore select a residue $r$ with $2r=r$ in the finite cyclic group, hence $r=0$. In particular, the multiples of every fixed positive integer belong to every additive idempotent ultrafilter.

For $A\in \mathcal{U}$ and an idempotent ultrafilter, put $A^{\ast }=\{n\in A:A-n\in \mathcal{U}\}$. Then $A^{\ast }\in \mathcal{U}$, and $A^{\ast }-n\in \mathcal{U}$ for every $n\in A^{\ast }$. For the second assertion, apply idempotence to $A-n$ and intersect its resulting good-translation set with $A-n$. This allows each new finite-sums generator to be chosen from finitely many translation constraints, proving the Idempotent-ultrafilter proof of Hindman's theorem.

An ultrafilter $\mathcal{U}\in \beta \mathbb{N}$ is idempotent when $\mathcal{U}+\mathcal{U}=\mathcal{U}$. The Ellis–Numakura lemma guarantees such an ultrafilter and supplies the main algebraic input to the Idempotent-ultrafilter proof of Hindman's theorem.