Affine bijections of ${C}({X},I)$

Janko Marovt ¹

¹ EPF - University of Maribor Razlagova 14 2000 Maribor, Slovenia

Studia Mathematica, Tome 173 (2006) no. 3, pp. 295-309

Résumé

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

Détail

DOI : 10.4064/sm173-3-4

Mots-clés : mathcal compact hausdorff space which satisfies first axiom countability mathcal mathcal set continuous functions mathcal varphi mathcal mathcal rightarrow mathcal mathcal bijective affine map there exists homeomorphism mathcal rightarrow every component mathcal have either varphi mathcal mathcal varphi f mathcal mathcal