Order preserving bijections of \(C_+(X)\)
Bijections that are order preserving in both directions from the space \(C^+(X)\) of all continuous functions from \(X\) to the nonnegative reals to itself are described. The author considered a similar situation [J. Math. Anal. Appl. 311, No.~2, 567--581 (2005; Zbl 1083.54010)] for the space \(C(X,I)\), all continuous maps from \(X\) to the \([0,1]\). Given the assumption that \(X\) is first countable and compact, then the central result of this paper establishes that if \(\varphi\) is a bijection such that \(\varphi\) and \(\varphi^{-1}\) are both order preserving, then \(\varphi\) is a modification of a translation, namely, \(\varphi(f)(x) = m_x(f(\omega(x)))\), where \(\omega\) from \(X\) to \(X\) is a homeomorphism and \(m_x\) is a bijective increasing map from the \([0,\infty)\) to itself. It is also noted that, as in the author's work cited above, it can be verified that given \(\omega\) is a homeomorphism and \(m_x\) an increasing bijection with a continuity condition related to \(m\), then the corresponding map \(\varphi(f)(x) = m_x(f(\omega(x))) \) is bijective and order preserving in both directions.
scientific article; zbMATH DE number 5788308
scientific article