Algebra in the Stone-Cech Compactification

U-filters and uniform compactification

“…We denote the minimal ideal by K. In what follows, we apply the fact that K is the union of all minimal left ideals of G LU C (see [HS,p. 34]).…”

A variant of the density Hales–Jewett theorem

“…With this operation, βW becomes a compact Hausdorff right topological semigroup; in particular, the map p → pq is continuous for each q ∈ βW . See [7] for full details.…”

Continuity inGLUC

“…The largest semigroup compactification of G with the joint continuity property, in the sense that any other is a natural quotient, is called the LU C-compactification and denoted by G LU C . The homomorphism ψ : G → G LU C is a topological embedding, so we identify G with its image, and write G * = G LU C \ G. As a topological compactification, G LU C is characterized by the property that a continuous function f : G → [0, 1] extends to a continuous functionf : G LU C → [0, 1] if and only if f is uniformly continuous with respect to the right uniformity (see [2,Theorem 21.41]). If G is locally compact, then every semigroup compactification of G has the joint continuity property (see [8,Theorem II.4.3]), so G LU C is the largest semigroup compactification of G. In the case, where G is discrete, G LU C coincides with βG, the Stone-Čech compactification.…”

“…Example 1.3. Let Z (2) denote the additive group of 2-adic integers and let H = ω Z (2) . Define the subgroups K and G of H by K = 2H = ω 2Z (2) and G = ω Z (2) + K. Define the group topology on G by taking the natural compact topology on K and declaring the subgroup K to be open.…”

“…Let Z (2) denote the additive group of 2-adic integers and let H = ω Z (2) . Define the subgroups K and G of H by K = 2H = ω 2Z (2) and G = ω Z (2) + K. Define the group topology on G by taking the natural compact topology on K and declaring the subgroup K to be open. Then G is a locally compact σ-compact torsion-free Abelian group, K an open compact subgroup of G, and N = G/K = ω Z 2 a countably infinite discrete Boolean group.…”