Quantum graph: Difference between revisions

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In [[mathematics]], two [[non-empty]] [[subset]]s ''A'' and ''B'' of a given [[metric space]] (''X'', ''d'') are said to be '''positively separated''' if the [[infimum]]
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:<math>\inf_{a \in A, b \in B} d(a, b)</math>
 
is strictly positive. (Some authors also specify that ''A'' and ''B'' should be [[disjoint sets]]; however, this adds nothing to the definition, since if ''A'' and ''B'' have some common point ''p'', then ''d''(''p'',&nbsp;''p'')&nbsp;=&nbsp;0, and so the infimum above is clearly 0 in that case.)
 
For example, on the real line with the usual distance, the [[open interval]]s (0,&nbsp;2) and (3,&nbsp;4) are positively separated, while (3,&nbsp;4) and (4,&nbsp;5) are not. In two dimensions, the graph of ''y''&nbsp;=&nbsp;1/''x'' for ''x''&nbsp;&gt;&nbsp;0 and the ''x''-axis are not positively separated.
 
==References==
 
* {{cite book
|    author = Rogers, C. A.
|    title = Hausdorff measures
|  edition = Third edition
|    series = Cambridge Mathematical Library
| publisher = Cambridge University Press
|  location = Cambridge
|      year = 1998
|    pages = xxx+195
|        id = ISBN 0-521-62491-6
}}
 
[[Category:Metric geometry]]

Revision as of 06:23, 20 February 2014

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