Monotone extenders for bounded $c$-valued functions

Kaori Yamazaki¹

¹ Faculty of Economics Takasaki City University of Economics 1300 Kaminamie, Takasaki Gunma 370-0801, Japan

Studia Mathematica, Tome 199 (2010) no. 1, pp. 17-22

Résumé

Let $c$ be the Banach space consisting of all convergent sequences of reals with the sup-norm, $C_\infty (A, c)$ the set of all bounded continuous functions $f:A\to c$, and $C_A(X, c)$ the set of all functions $f:X\to c$ which are continuous at each point of $A \subset X$. We show that a Tikhonov subspace $A$ of a topological space $X$ is strong Choquet in $X$ if there exists a monotone extender $u: C_\infty (A, c)\to C_A(X, c)$. This shows that the monotone extension property for bounded $c$-valued functions can fail in GO-spaces, which provides a negative answer to a question posed by I. Banakh, T. Banakh and K. Yamazaki.

Détail

DOI : 10.4064/sm199-1-2

Keywords: banach space consisting convergent sequences reals sup norm infty set bounded continuous functions set functions which continuous each point subset tikhonov subspace topological space strong choquet there exists monotone extender infty shows monotone extension property bounded c valued functions fail go spaces which provides negative answer question posed banakh banakh yamazaki