A short proof of completion theorem for metric spaces
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 13 (2021) no. 2, pp. 61-64
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The completion theorem for metric spaces is always proven using the space of Cauchy sequences. In this paper, we give a short and alternative proof of this theorem via Zorn's lemma. First, we give a way of adding one point to an incomplete space to get a chosen non-convergent Cauchy sequence convergent. Later, we show that every metric space has a completion by constructing a partial ordered set of metric spaces.
Keywords:
Completion theorem, metric space, complete space, Zorn's lemma.
@article{VYURM_2021_13_2_a8,
author = {U. Kaya},
title = {A short proof of completion theorem for metric spaces},
journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
pages = {61--64},
publisher = {mathdoc},
volume = {13},
number = {2},
year = {2021},
language = {en},
url = {http://geodesic.mathdoc.fr/item/VYURM_2021_13_2_a8/}
}
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%0 Journal Article %A U. Kaya %T A short proof of completion theorem for metric spaces %J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika %D 2021 %P 61-64 %V 13 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/VYURM_2021_13_2_a8/ %G en %F VYURM_2021_13_2_a8
U. Kaya. A short proof of completion theorem for metric spaces. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 13 (2021) no. 2, pp. 61-64. http://geodesic.mathdoc.fr/item/VYURM_2021_13_2_a8/