Positively separated sets
- Positively separated sets
In mathematics, two non-empty subsets "A" and "B" of a given metric space ("X", "d") are said to be positively separated if the infimum
:
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", "p") = 0, and so the infimum above is clearly 0 in that case.)
For example, on the real line with the usual distance, the open intervals (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph of "y" = 1/"x" for "x" > 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
Wikimedia Foundation.
2010.
Look at other dictionaries:
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
Metric outer measure — In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that for every pair of positively separated subsets A and B of X. Construction of metric outer measures Let… … Wikipedia
cell — cell1 cell like, adj. /sel/, n. 1. a small room, as in a convent or prison. 2. any of various small compartments or bounded areas forming part of a whole. 3. a small group acting as a unit within a larger organization: a local cell of the… … Universalium
chemical bonding — ▪ chemistry Introduction any of the interactions that account for the association of atoms into molecules, ions, crystals, and other stable species that make up the familiar substances of the everyday world. When atoms approach one another … Universalium
Christianity — /kris chee an i tee/, n., pl. Christianities. 1. the Christian religion, including the Catholic, Protestant, and Eastern Orthodox churches. 2. Christian beliefs or practices; Christian quality or character: Christianity mixed with pagan elements; … Universalium
arts, East Asian — Introduction music and visual and performing arts of China, Korea, and Japan. The literatures of these countries are covered in the articles Chinese literature, Korean literature, and Japanese literature. Some studies of East Asia… … Universalium
performing arts — arts or skills that require public performance, as acting, singing, or dancing. [1945 50] * * * ▪ 2009 Introduction Music Classical. The last vestiges of the Cold War seemed to thaw for a moment on Feb. 26, 2008, when the unfamiliar strains … Universalium
radio — /ray dee oh /, n., pl. radios, adj., v., radioed, radioing. n. 1. wireless telegraphy or telephony: speeches broadcast by radio. 2. an apparatus for receiving or transmitting radio broadcasts. 3. a message transmitted by radio. adj. 4. pertaining … Universalium
spectroscopy — spectroscopist /spek tros keuh pist/, n. /spek tros keuh pee, spek treuh skoh pee/, n. the science that deals with the use of the spectroscope and with spectrum analysis. [1865 70; SPECTRO + SCOPY] * * * Branch of analysis devoted to identifying… … Universalium
Life Sciences — ▪ 2009 Introduction Zoology In 2008 several zoological studies provided new insights into how species life history traits (such as the timing of reproduction or the length of life of adult individuals) are derived in part as responses to… … Universalium
