ABSTRACT
Let X be a locally compact Hausdorff space, c 0 ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1220.tif"/> the continuous complex functions vanishing at infinity on X.
If F1,F2 are disjoint compact subsets of X and ϵ ≥ 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1221.tif"/> then a function f ϵ c 0 ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1222.tif"/> such that | f ( F 1 ) | ≤ ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1223.tif"/> and | 1 − f ( F 2 ) | ≤ ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1224.tif"/> is called an ϵ-idempotent with respect to the pair (F1,F2).
Call E ⊂ c 0 ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1225.tif"/> regular if E contains a 0-idempotent with respect to each pair ( { x } , F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1226.tif"/> with F compact and x x ∉ F . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1227.tif"/>
Call E ⊂ c 0 ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1228.tif"/> ϵ-normal if E contains an e-idempotent with respect to each pair of disjoint compacta in X and call E simply normal if it is 0-normal.
If E is a normed linear space lying in c0(X), call E boundedly ϵ-normal if there is a constant K such that the ϵ-idempotents required in (iii) can be found with E-norm less than K.
If E is a normed linear space lying in c0(X), call E locally boundedly ϵ-normal if X is a union of open sets U for which there exist constants KU such that E contains an ϵ-idempotent of norm less than KU for every pair of disjoint compacta in U ¯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072645/d0c716e9-d02e-4ae2-8880-b8eca47a1243/content/eq1229.tif"/> .
