Counting linearly ordered spaces

Gerald Kuba

Colloquium Mathematicae, Tome 135 (2014), p. 1-14 / Harvested from The Polish Digital Mathematics Library

Résumé

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).

Détail

Publié le : 2014-01-01

Zbl 1300.54042

EUDML-ID : urn:eudml:doc:284343