What exactly is separability or countable order - density - as opposed to density as in a dense or bottom-less total order. particularly with regard to probabilistic representations?
I know that its a requirement for a real valued order- preserving function.
Incidentally I presume that when they use bi-conditionals these can be replaced by strict bi-conditionals, that they mean order preserving and reflecting, an embedding or a strictly mon0tone increasing functions. Is there any reason unless the domain is non numerical in certain cases for the use of x
Please let me know if the following references are helpful:
-----> 1. Topology and its Applications 159 (2012) 3103–3114
Contents lists available at SciVerse ScienceDirect
Topology and its Applications
www.elsevier.com/locate/topol
The embedding structure for linearly ordered topological spaces
A. Primavesi a,1, K. Thompson b,∗,2
a School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
b Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, A-1040 Wien, Austria
a r t i c l e i n f o a b s t r a c t
MSC:
primary 54F05
secondary 06F30, 03E75
Keywords:
Linearly ordered topological space
LOTS
Universal structure
Basis
Well-quasi-order
In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality κ where κ 2ℵ0 , it is proved that this quasi-order has no maximal element and that in fact the dominating num r for such quasi-orders is maximal, i.e. 2κ . Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal structure of their respective embedding quasi-order. The basis problem for uncountable LOTS is also considered; assuming the Proper Forcing Axiom there is an eleven element basis for the class of uncountable LOTS and a six element basis for the class of dense uncountable LOTS in which all points have countable cofinality and coinitiality.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The embedding structure for linear orders is dependent on the axiomatisation of set theory that we choose to adopt; this is a widely known example of set-theoretic independence phenomena occurring at the heart of classical mathematics.
For example, there can consistently be a universal linear order (one that embeds every other linear order of the same cardinality) for every infinite cardinal κ. This statement can also consistently fail to hold. More recently, it has been shown
that in certain models of set theory there is a five element basis for the uncountable linear orders, resolving a long-standing open question; other models of set theory do not admit a finite basis for the uncountable linear orders In this paper we investigate the properties of the embedding structure for linear orders with extra topological structure and compare them to the properties of the embedding structure for linear orders. A linearly ordered topological space (or LOTS) is a linear order endowed with the open interval topology, call it τ. An embedding is an injective function that preserves structure. Specifically, for the structures in this paper • A linear order embedding is an injective order-preserving map.• A LOTS embedding is an injective order-preserving map that is continuous. The existence of a LOTS embedding, f : A→ B, where A and B are two arbitrary linear orders, ensures that not only is
there a suborder of B, call it B , that is order-isomorphic to A, but also that the open sets in τB are exactly those sets of the form B ∩ u for some u ∈ τB . The existence of a LOTS embedding is a non-trivial, natural strengthening of the existence of a linear order embedding.
* Corresponding author.
E-mail addresses: [email protected] (A. Primavesi), [email protected] (K. Thompson).
1 Part of this work was done while the first author was a visiting scholar at the Technischen Universität Wien.
2 The second author was supported by Elise-Richter project number V142-N13 from the FWF (Austrian Science Fund).
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
-----> (C) The rationals and reals as linearly ordered
site: sets. http://www.cs.man.ac.uk/~hsimmons/SetTheory/C-QandR.pdf
-----> Separability
Author: Keith Conrad
site: http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/separable1.pdf
Dennis
Dennis Mazur
As in the article presented the topic is about Probability Spaces, and in these is a topology, there are criteria for the study of continuity that are closely related to separability, compactness, density and others. I leave a great book utility. (two versions file pdf and djvu)
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