Affine bijections of ${C}({X},I)$
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 $\varphi (f)(x)=1-f(\mu(x))$, $f\in \mathcal{C}(\mathcal{X},I)$, $x\in C$.
