Homogeneous linearly ordered spaces
Abstract
Every compact subset of a homogeneous generalized ordered (GO) space has character at most ω1 and cardinality at most 2ω1; if such a subset has uncountable character, then the character of the whole space equals ω1 and its π-character is countable. We construct a homogeneous σ-compact linearly ordered space (LOTS) H containing a compact subset S of cardinality 2ω1 whose character is ω1 at every point and whose weight and Souslin number are both 2ω1; thus both bounds obtained are sharp. We prove that a semitopological group that is a GO space is hereditarily paracompact; if, in addition, it is not a P-space, then it is submetrizable, has countable character, and its compact subsets are metrizable. Every linearly ordered semitopological group (and, more generally, every GO semitopological group) is either metrizable or is a P-space; the same holds for topological groups. We also show that in an order-homogeneous LOTS every compact subset is first countable.
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BibTeX
@misc{lipin2026homogeneous,
title = {{Homogeneous linearly ordered spaces}},
author = {Anton Lipin and Evgenii Reznichenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15499},
eprint = {2609.15499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15499},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homogeneous linearly ordered spaces
AU - Anton Lipin
AU - Evgenii Reznichenko
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15499
UR - https://arxiv.org/abs/2609.15499
AB - Every compact subset of a homogeneous generalized ordered (GO) space has character at most ω_1 and cardinality at most 2^ω_1; if such a subset has uncountable character, then the character of the whole space equals ω_1 and its π-character is countable. We construct a homogeneous σ-compact linearly ordered space (LOTS) H containing a compact subset S of cardinality 2^ω_1 whose character is ω_1 at every point and whose weight and Souslin number are both 2^ω_1; thus both bounds obtained are sharp. We prove that a semitopological group that is a GO space is hereditarily paracompact; if, in addition, it is not a P-space, then it is submetrizable, has countable character, and its compact subsets are metrizable. Every linearly ordered semitopological group (and, more generally, every GO semitopological group) is either metrizable or is a P-space; the same holds for topological groups. We also show that in an order-homogeneous LOTS every compact subset is first countable.
ER -