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