A net in a Hausdorff uniform space is called cofinally Cauchy if for each entourage, there exists a cofinal (rather than residual) set of indices whose corresponding terms are pairwise within the entourage. In a metric space equipped with the associated metric uniformity, if each cofinally Cauchy sequence has a cluster point, then so does each cofinally Cauchy net, and the space is called cofinally complete. Here we give necessary and sufficient conditions for the nonempty closed subsets of the metric space equipped with Hausdorff distance to be cofinally complete.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-5,
author = {Gerald Beer and Giuseppe Di Maio},
title = {Cofinal completeness of the Hausdorff metric topology},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {75-85},
zbl = {1198.54033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-5}
}
Gerald Beer; Giuseppe Di Maio. Cofinal completeness of the Hausdorff metric topology. Fundamenta Mathematicae, Tome 209 (2010) pp. 75-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-5/