Algebraic Structures and Compactness. A Review of the Most Important Results

Abstract

A fundamental connection between algebraic structures and compacta is realized through the concept of an ideal. This connection is most clearly visible in the case of mutual correspondence between an arbitrary Tikhonov space X and the ring C⁰ (X) of all bounded, continuous real-valued functions on X. Denote by H⁰(X) the set of all nontrivial homomorphisms of the ring C⁰ (X) into the ring ℝ of real numbers and introduce the topology of pointwise convergence in H⁰(X) To each x ∈ X there corresponds the evaluation homomorphism ω _x: C⁰ (X) → ℝ described by ω _x (f) = f(x), for all f ∈ C ⁰(X). The mapping of X into the space H ⁰(X) obtained in this way is denoted by µ Every homomorphism ω ∈ H ⁰(X) is onto, that is ω(C⁰(X)) =ℝ, and its kernel ker(ω) is a maximal ideal of C ⁰ (X). Conversely, to every maximal ideal of the ring C⁰ (X) there corresponds a homomorphism of C⁰ (X) onto ℝ, that is an element of the set H ⁰(X). Thus the set H ⁰(X) can be canonically identified with the set of all maximal ideals of the ring C⁰(X). The mapping μ of the space X into the space H⁰(X) described by x → ω _x is a homeomorphism of X onto a subspace of the space H⁰ (X). The following result is known as the Gel’fand-Kolmogorov theorem (see Naimark (1956)).