Spaces of maximal points

1997 ◽  
Vol 7 (5) ◽  
pp. 543-555 ◽  
Author(s):  
JIMMIE LAWSON
Keyword(s):  
Metric Spaces ◽  
Scott Topology ◽  
Maximal Points

This paper shows that it is precisely the complete metrizable separable metric spaces that can be realized as the set of maximal points of an ω-continuous dcpo, where the set of maximal points is topologized with the relative Scott topology.

scholarly journals Dcpo models of T1 spaces

10.29007/prcv ◽  
2018 ◽  
Author(s):  
Zhao Dongsheng ◽  
Xi Xiaoyong
Keyword(s):  
Topological Space ◽  
Positive Answer ◽  
Scott Topology ◽  
Maximal Points

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP consisting of all maximal points of P is homeomorphic to X. Every T<sub>1</sub> space has a (bounded complete algebraic) poset model. It is, however, not known whether every T<sub>1</sub> space has a dcpo model and whether every sober T<sub>1</sub> space has a dcpo model whose Scott topology is sober. In this paper we give a positive answer to these two problems. For each T<sub>1</sub> space X we shall construct a dcpo A that is a model of X, and prove that X is sober if and only if the Scott topology of A is sober. One useful by-product is a method that can be used to construct more non-sober dcpos.

A quantitative computational model for complete partial metric spaces via formal balls

2009 ◽  
Vol 19 (3) ◽  
pp. 541-563 ◽  
Author(s):  
SALVADOR ROMAGUERA ◽  
OSCAR VALERO
Keyword(s):  
Computational Model ◽  
Partial Order ◽  
Metric Space ◽  
Metric Spaces ◽  
Partial Metric Space ◽  
Partial Metric ◽  
Scott Topology ◽  
Partial Metric Spaces

Given a partial metric space (X, p), we use (BX, ⊑dp) to denote the poset of formal balls of the associated quasi-metric space (X, dp). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑dp). In particular, we prove that a partial metric space (X, p) is complete if and only if the poset (BX, ⊑dp) is a domain. Furthermore, for any complete partial metric space (X, p), we construct a Smyth complete quasi-metric q on BX that extends the quasi-metric dp such that both the Scott topology and the partial order ⊑dp are induced by q. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Künzi, H. Pajoohesh and M. P. Schellekens (Künzi et al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.

Directed complete poset models of T1 spaces

2016 ◽  
Vol 164 (1) ◽  
pp. 125-134 ◽  
Author(s):  
DONGSHENG ZHAO ◽  
XIAOYONG XI
Keyword(s):  
Topological Space ◽  
Positive Answer ◽  
Scott Topology ◽  
Maximal Points

AbstractA poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space Σ E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

Some Common Fixed Point Theorem in Complete Metric Space of Integral Type

2018 ◽  
Vol 6 (8) ◽  
pp. 297
Author(s):  
Jagdish C. Chaudhary ◽  
Shailesh T. Patel
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Contractive Condition ◽  
Integral Type ◽  
Complete Metric Spaces ◽  
Common Fixed Point Theorems ◽  
Common Fixed Point Theorem

In this paper, we prove some common fixed point theorems in complete metric spaces for self mapping satisfying a contractive condition of Integral  type.

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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