scholarly journals Fuzzy quasi-metrics for the Sorgenfrey line

2013 ◽  
Vol 222 ◽  
pp. 98-107 ◽  
Author(s):  
Valentín Gregori ◽  
Samuel Morillas ◽  
Bernardino Roig
Keyword(s):  
Sorgenfrey Line

scholarly journals Subspaces of the Sorgenfrey Line

1998 ◽  
Vol 90 (1-3) ◽  
pp. 57-68 ◽  
Author(s):  
Dennis K. Burke ◽  
J.Tatch Moore
Keyword(s):  
Sorgenfrey Line

Maps of the Sorgenfrey line

2011 ◽  
Vol 272 (S1) ◽  
pp. 287-291
Author(s):  
N. V. Velichko
Keyword(s):  
Sorgenfrey Line

The Sorgenfrey line is non-metrizable

2011 ◽  
Vol 133 (1-2) ◽  
pp. 185-187
Author(s):  
M. Bognár
Keyword(s):  
Sorgenfrey Line

scholarly journals Subspaces of the Sorgenfrey line and their products

2006 ◽  
Vol 30 (2) ◽  
pp. 401-413
Author(s):  
Vitalij A. Chatyrko
Keyword(s):  
Sorgenfrey Line

scholarly journals Separating minimal valuations, point-continuous valuations, and continuous valuations

2021 ◽  
pp. 1-19
Author(s):  
Jean Goubault-Larrecq ◽  
Xiaodong Jia
Keyword(s):  
Lebesgue Measure ◽  
List Type ◽  
Type Number ◽  
Scott Topology ◽  
Sorgenfrey Line ◽  
Continuous Valuation

Abstract We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations: (1) Let ${\mathcal J}$ be the Johnstone’s non-sober dcpo, and μ be the continuous valuation on ${\mathcal J}$ with μ(U)=1 for nonempty Scott opens U and μ(U)=0 for $U=\emptyset$ . Then, μ is a point-continuous valuation on ${\mathcal J}$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb{R}_\ell$ . Its restriction to the open subsets of $\mathbb{R}_\ell$ is a continuous valuation λ. Then, its image valuation $\overline\lambda$ through the embedding of $\mathbb{R}_\ell$ into its Smyth powerdomain $\mathcal{Q}\mathbb{R}_\ell$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $\overline\lambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo’s.

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