On the ideal structure of 𝐶(𝑋)

The ideal structure of C ( X ) C(X) , the algebra of continuous functions from a completely regular Hausdorff space X X to the scalars is analyzed by examining for fixed A ⊂ β X A ⊂ β X (the Stone-Čech compactification of X X ) the structure of the quotient I A / F A {I^A}/{F^A} , where I A [ F A ] {I^A}[{F^A}] is the ideal of maps f ∈ C ( X ) f ∈ C(X) for which \[ A ⊂ cl β X Z ( f ) [ A ⊂ int β X cl β X Z ( f ) ] . A ⊂ {cl β X}Z(f) [A ⊂ {int β X}{cl β X}Z(f)].