On the relativization of chain topologies

authored by: Marcel Erné
Abstract: The intrinsic topology s_≤of a chain (X, ≤) induces on any subchain Y⊂X the relative topology s_≤Y. On the other hand, any such subchain Y is endowed with its own intrinsic topology s_≤y. We establish several necessary and sufficient conditions under which both topologies coincide, by suitably weakening the properties of convexity (Lemma 2), order-density (Theorem 3) and subcompleteness (Theorem 4), respectively. Another necessary and sufficient condition for the equation s_≤y = s_≤yformulated in terms of cuts, is given in Theorem 2. Besides other related results, we find a purely order-theoretical characterization of those subchains which are compact (Lemma 1) or connected (Corollary 2), respectively, in the intrinsic topology of the entire chain. As a simple consequence of Theorem 4, we obtain the wellknown result that the intrinsic topology of a chain can be obtained by relativization from the intrinsic topology of the normal completion (Corollary 9). We conclude with several applications to the Euclidean topology on R.
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
Type: Article
Journal: Pacific journal of mathematics
Volume: 84
Pages: 43-52
No. of pages: 10
ISSN: 0030-8730
Publication date: 01.09.1979
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Mathematics
Electronic version(s): https://doi.org/10.2140/pjm.1979.84.43 (Access: Open)

BibTeX

@article{b3b6c7319d1a43728f61f1c34f41699f,
title = "On the relativization of chain topologies",
abstract = "The intrinsic topology s≤of a chain (X, ≤) induces on any subchain Y⊂X the relative topology s≤Y. On the other hand, any such subchain Y is endowed with its own intrinsic topology s≤y. We establish several necessary and sufficient conditions under which both topologies coincide, by suitably weakening the properties of convexity (Lemma 2), order-density (Theorem 3) and subcompleteness (Theorem 4), respectively. Another necessary and sufficient condition for the equation s≤y = s≤yformulated in terms of cuts, is given in Theorem 2. Besides other related results, we find a purely order-theoretical characterization of those subchains which are compact (Lemma 1) or connected (Corollary 2), respectively, in the intrinsic topology of the entire chain. As a simple consequence of Theorem 4, we obtain the wellknown result that the intrinsic topology of a chain can be obtained by relativization from the intrinsic topology of the normal completion (Corollary 9). We conclude with several applications to the Euclidean topology on R.",
author = "Marcel Ern{\'e}",
year = "1979",
month = sep,
day = "1",
doi = "10.2140/pjm.1979.84.43",
language = "English",
volume = "84",
pages = "43--52",
journal = "Pacific journal of mathematics",
issn = "0030-8730",
publisher = "University of California, Berkeley",
number = "1",
}