| Popis: |
Let X be a Tychonoff space. Associated with every subset S of βX are the ideals M S = { f ∈ C ( X ) | S ⊆ cl β X Z ( f ) } and O S = { f ∈ C ( X ) | S ⊆ int β X cl β X Z ( f ) } of the ring C ( X ) , where Z ( f ) denotes the zero-set of f. We show that 〈 C ( f ) [ O K ] 〉 = O ( β f ) − 1 [ K ] for any continuous map f : X → Y and every closed subset K of βY, where β f : β X → β Y is the Stone extension of f and C ( f ) : C ( Y ) → C ( X ) is the ring homomorphism g ↦ g ∘ f . On the other hand, C ( f ) − 1 [ M S ] = M ( β f ) [ S ] for every subset S of βX if and only if f is a WN-map, in the sense of Woods [12] . These results (and others) are corollaries of more general ones obtained in pointfree function rings. |
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