Let 𝒳 be a compact Hausdorff space which satisfies the first axiom of countability, I = [0,1] and 𝓒(𝒳,I) the set of all continuous functions from 𝒳 to I. If φ: 𝓒(𝒳,I) → 𝓒(𝒳,I) is a bijective affine map then there exists a homeomorphism μ: 𝒳 → 𝒳 such that for every component C in 𝒳 we have either φ(f)(x) = f(μ(x)), f ∈ 𝓒(𝒳,I), x ∈ C, or φ(f)(x) = 1-f(μ(x)), f ∈ 𝓒(𝒳,I), x ∈ C.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-3-4,
author = {Janko Marovt},
title = {Affine bijections of C(X,I)},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {295-309},
zbl = {1101.46033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-3-4}
}
Janko Marovt. Affine bijections of C(X,I). Studia Mathematica, Tome 173 (2006) pp. 295-309. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-3-4/