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## Colloquium Mathematicum

2014 | 135 | 1 | 1-14
Tytuł artykułu

### Counting linearly ordered spaces

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EN
For a transfinite cardinal κ and i ∈ {0,1,2} let $ℒ_i(κ)$ be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if $κ < 2^{ℵ₀}$, and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The class ℒ₁(κ) ∩ ℒ₂(κ) contains precisely $2^{κ}$ spaces up to homeomorphism for every $κ ≥ 2^{ℵ₀}$. Our main results are explicit constructions which prove that both classes ℒ₀(κ) ∩ ℒ₁(κ) and ℒ₀(κ) ∩ ℒ₂(κ) contain precisely $2^{κ}$ spaces up to homeomorphism for every κ > ℵ₀. Moreover, for any κ we investigate the variety of second countable spaces in the class ℒ₀(κ) ∩ ℒ₁(κ) and the variety of first countable spaces of arbitrary weight in the class ℒ₂(κ).
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Rocznik
Tom
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1-14
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wydano
2014
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autor
• Institute of Mathematics, University of Natural Resources and Life Sciences, 1180 Wien, Austria
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