We study the Gromov-Hausdorff and Kadets distances between C(K)-spaces and their quotients. We prove that if the Gromov-Hausdorff distance between C(K) and C(L) is less than 1/16 then K and L are homeomorphic. If the Kadets distance is less than one, and K and L are metrizable, then C(K) and C(L) are linearly isomorphic. For K and L countable, if C(L) has a subquotient which is close enough to C(K) in the Gromov-Hausdorff sense then K is homeomorphic to a clopen subset of L.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-2-4,
author = {Yves Dutrieux and Nigel J. Kalton},
title = {Perturbations of isometries between C(K)-spaces},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {181-197},
zbl = {1081.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-2-4}
}
Yves Dutrieux; Nigel J. Kalton. Perturbations of isometries between C(K)-spaces. Studia Mathematica, Tome 166 (2005) pp. 181-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-2-4/