Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika"
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Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika", 2021, Volume 13, Issue 2, Pages 61–64
DOI: https://doi.org/10.14529/mmph210209
(Mi vyurm483)
 

Short communications

A short proof of completion theorem for metric spaces

U. Kaya

Bitlis Eren University, Bitlis, Turkey
References:
Abstract: 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.
Received: 30.01.2021
Document Type: Article
UDC: 515.124
MSC: 54E50, 54A20, 06A06
Language: English
Citation: U. Kaya, “A short proof of completion theorem for metric spaces”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 13:2 (2021), 61–64
Citation in format AMSBIB
\Bibitem{Kay21}
\by U.~Kaya
\paper A short proof of completion theorem for metric spaces
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2021
\vol 13
\issue 2
\pages 61--64
\mathnet{http://mi.mathnet.ru/vyurm483}
\crossref{https://doi.org/10.14529/mmph210209}
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