1. CMB 2010 (vol 53 pp. 719)
|A Continuous Extension Operator for Convex Metrics|
We consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.
Categories:54E35, 54C20, 54E40
2. CMB 2001 (vol 44 pp. 266)
|Extension of Maps to Nilpotent Spaces |
We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$.
Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space
Categories:55M10, 55S36, 54C20, 54F45
3. CMB 2000 (vol 43 pp. 208)
|Extensions of Continuous and Lipschitz Functions |
We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant. As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.
Keywords:extension, continous, Lipschitz, Banach space