Rings of Continuous Functions

Abstract

Let X be a nonvoid set. A topology on X is a family τ⊂ P(X) of subsets which is closed under finite intersections and arbitrary unions. The pair (X,τ) is called a topological space, the elements ofτare called the open sets of the space. If τ₁ and τ₂ are topologies on X and τ₁ ⊂ τ₂ we say that τ₂ is finer than τ₁ or that τ₁ is coarser than τ₂. It exists in X a finest topology τ° = P(X) (called the discrete topology) and a coarsest one τ₀ = {θ, X} (called the indiscrete topology). The elements of F={X \ G|G ∈ τ} are called the closed sets.