|
Short communications
A short proof of completion theorem for metric spaces
U. Kaya Bitlis Eren University, Bitlis, Turkey
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
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
Linking options:
https://www.mathnet.ru/eng/vyurm483 https://www.mathnet.ru/eng/vyurm/v13/i2/p61
|
Statistics & downloads: |
Abstract page: | 70 | Full-text PDF : | 45 | References: | 14 |
|