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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao zbMATH Openarrow_drop_down
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
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Metric for metrizable GO-spaces

Authors: Bennett, H. R.;

Metric for metrizable GO-spaces

Abstract

The mathematical community is notorious for knowing that a solution to a problem exists without actually knowing how to obtain a solution. The metrizability of GO-spaces is a case in point. Let X be a GO-space constructed on a LOTS (Y,\(\tau)\), let \(I=\{x\in X:\{x\}\in \tau \}\), \(R=\{x\in X:[x,\to)\in \tau \}\), \(L=\{x\in X:(\leftarrow,x]\in \tau \}\) and let \(D=I\cup R\cup L\). It is known that a GO-space is metrizable if and only if D is \(\sigma\)-discrete. The author constructs a metric for a countable GO-space, which, if D is dense, depends only on the enumeration of the points of X and the ordering of X. More generally, assuming that the \(\sigma\)-discrete subsets I, R, and L of a metrizable GO-space are realized as countable nested unions of discrete sets \(I=\cup_{n\in N}In\), \(R=\cup_{n\in N}Rn\), and \(L=\cup_{n\in N}Ln\), the author constructs a metric, which, if D is dense, depends only on the ordering of X and the intersection of intervals with members of \(\{Rn\}_{n\in N,}\{Ln\}_{n\in N}\) and \(\{In\}_{n\in N}\). The construction of the metrics is well motivated and well illustrated, and, although the use of both \(\emptyset\) and \(\Phi\) to denote the empty set is unfortunate, the paper on the whole is easy to follow.

Keywords

metrizability of GO-spaces, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, Metric spaces, metrizability

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