# Isolated Point

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## isolated point

[′ī·sə‚lād·əd ′point]
(mathematics)
A point p in a topological space is an isolated point of a set if p is in the set and there is a neighborhood of p which contains no other points of the set.
A point that satisfies the equation for a plane curve C but has a neighborhood that includes no other point of C. Also known as acnode; hermit point.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Isolated Point

a point belonging to some set M in whose neighborhood there is no other point of the set. The points of the set M that do not satisfy this condition are its limit points. The definition of an isolated point given above presupposes that in set Mthe concept of proximity among its elements (points) has been introduced. Because of this, the concept of isolated point is topological. In particular, if M is a set of points on a straight line, then point x of this set is an isolated point if there exists an interval that contains this point and does not contain other points of set M. Thus, if M consists of points having coordinates 1, 1/2, 1/3, …, l/n, …, then each point of this set is an isolated point, but for a set consisting of the same points and of a point having the coordinate 0, the latter will not be an isolated point. The isolated points of a curve or surface (here M is the set of all points of the given curve or surface) are also considered in geometry; for example, the point (0, 0) is an isolated point of the curve y2 = x4 − 4 x2. In the theory of functions of a complex variable we speak of the isolated singular points of an analytic function; the pole of a single-valued analytic function may serve as an example of this.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
However, an isolated point concentrates all other points on the opposite end of the region of interest, resulting in a considerable number of empty strips and in the increased population density of those few strips containing the entire point cloud (Fig.
The CRG algorithm can be used to obtain the K connected regions and an isolated point set.
Then, we obtained 82 complete car-following processes that included 3,924 isolated point data.
But then [lambda] [member of] [[PHI].sub.sf] (A) [product] [sigma](A) is (an isolated point of [sigma](A) which happens to be) a finite rank pole of A.
The total perimeter, including an isolated point and a certain region, is bound to be bigger than the region's perimeter, as shown in Figures 1(b) and 1(d).
Definition 11  Let A be a subset of a space (Eq.) Then a point (Eq.) is said to be a semi isolated point of (Eq.) , if there is a semi open set such that (Eq.).
The digitally literate clients of today now effortlessly utilise the best sources of online intelligence to track agencies and work on a daily basis - not simply at the isolated point of searching for a new partner.
In this paper, we prove that [k.sup.*]-paranormal operators have H property and if 0 [not equal to] A is an isolated point of the spectrum of [k.sup.*]-paranormal operator T for a positive integer k, then the Riesz idempotent operator E of T with respect to [lambda] satisfies [E.sub.[lambda]]H = ker(T - [lambda]) = ker[(T - [lambda]).sup.*].
In one of the worst cases of river pollution, the sugar giant is discharging its effluents through a drain that opens into the mainstream river at an isolated point at Puth village, about 30 km from Garhmukteswar ( Brijghat) in the Ghaziabad district.
We want someone capable of seeing a big, complicated, important relationship that will affect American interests for decades to come--not someone who seizes on one isolated point or sticks to a slogan.
Still, the book's authors find lots to criticize, including Shanghai's "decision to accommodate growth by building high, with isolated point blocks surrounded by car ramps and empty open space, damaging the subtle urban grain of a city of immense character and dynamic street life." Also faulted: forced relocation of innercity residents to remote housing projects, and the banning (gasp!) of bicycles on certain streets, supposedly to relieve congestion.

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