Recent Developments in the Topology of Ordered Spaces

Bennett, Harold; Lutzer, David

doi:10.1016/b978-044450980-2/50003-0

Cited by 13 publications

(11 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence x is the maximum of A. If x ∈ B, then, by (5) and since A ∪ B = X, (−∞, x] is a neighborhood of x, hence, by D-convergence, {i ∈ I | x i ≤ x} ∈ D, and this contradicts again x ∈ A. Hence x ∈ B.…”

Section: D-convergence Of a Given Sequencementioning

confidence: 78%

Ultrafilter Convergence in Ordered Topological Spaces

Lipparini

2015

Order

View full text Add to dashboard Cite

We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter D, the notions of D-compactness and of D-pseudocompactness are equivalent. Any product of initially λ-compact generalized ordered topological spaces is still initially λ-compact. On the other hand, preservation under products of certain compactness properties is independent from the usual axioms for set theory

show abstract

Section: D-convergence Of a Given Sequencementioning

confidence: 78%

Ultrafilter Convergence in Ordered Topological Spaces

Lipparini

2015

Order

View full text Add to dashboard Cite

show abstract

“…Then there exists a continuous order-preserving f : X → [0, 1] such that A ⊆ f −1 (1) and B ⊆ f −1 (0). Without loss of generality we may assume that (1) is a clopen upset separating A and B. Else, there exists x ∈ X − ( f −1 (1) ∪ f −1 (0)) which is an upper bound of f −1 (0) and a lower bound of f −1 (1). If x is the minimum of X − f −1 (0), then by Lemma 1.4, ↑x is a clopen upset of X, containing A and missing B.…”

Section: Lemma 33 If X Is a Zero-dimensional Totally Ordered Space mentioning

confidence: 99%

“…As h is order-preserving and Y is totally ordered, this means that g(D, 0) < g(D, 1), and so D ∈ S Y . Thus, 1), which is equivalent to π S (D, 0) = π S (D, 1). But this means that D / ∈ S X/∼ S .…”

Section: Theorem 213 the Maps ϕ And ψ Establish An Order-isomorphismmentioning

confidence: 99%

Order-Compactifications of Totally Ordered Spaces: Revisited

Bezhanishvili

Morandi

2011

Order

View full text Add to dashboard Cite

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56-65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683-694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011-1014, 1966 Sib Math J 10:124-132, 1969) and Kaufman (Colloq Math 17:35-39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13: [56][57][58][59][60][61][62][63][64][65] 1975) and Kent and Richmond (J Math Math Sci 11(4):683-694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.

show abstract

“…The most important open question in GO-space theory is Maurice's problem, which Qiao and Tall showed [19] is closely related to several other old questions of Heath and Nyikos [5]. Maurice asked whether there is a ZFC example of a perfect GO-space that does not have a σ-closed-discrete dense subset.…”

Section: A Few Of Our Favorite Thingsmentioning

confidence: 99%

“…For many of the questions, space limitations restricted us to giving only definitions and references for the question. For more detail, see [5]. Notably absent from our list are problems about orderability, about products of special ordered spaces, about continuous selections of various kinds, and about C p -theory, and for that we apologize.…”

Section: Introductionmentioning

confidence: 99%