Affine bijections of
Tom 173 / 2006
Studia Mathematica 173 (2006), 295-309
MSC: Primary 46J10; Secondary 46E05.
DOI: 10.4064/sm173-3-4
Streszczenie
Let \mathcal{X} be a compact Hausdorff space which satisfies the first axiom of countability, I=[ 0,1] and \mathcal{C}(\mathcal{X}, I) the set of all continuous functions from \mathcal{X} to I. If \varphi:\mathcal{C}(\mathcal{X},I) \rightarrow\mathcal{C}(\mathcal{X},I) is a bijective affine map then there exists a homeomorphism \mu:\mathcal{X\rightarrow X} such that for every component C in \mathcal{X} we have either \varphi (f)(x)=f(\mu(x)), f\in \mathcal{C}(\mathcal{X},I), x\in C , or \varphi (f)(x)=1-f(\mu(x)), f\in \mathcal{C}(\mathcal{X},I), x\in C.