Limit Point

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limit point

[′lim·ət ‚pȯint]
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 It might be outdated or ideologically biased.

Limit Point


(or accumulation point). A limit point of a set A in a metric space is a point ξ in a space such that arbitrarily close to ξ there are points in A distinct from ξ. In other words, ξ is a limit point if any neighborhood of ξ contains an infinite number of points in A. A characteristic property of ξ is the existence of at least one sequence of distinct points of A that converges to ξ.

A limit point of a set does not have to belong to the set. Thus, every point on the real axis is a limit point for the set of rational points, because for every number—rational or irrational—we can find a sequence of distinct rational numbers that converges to it. Not every infinite set has a limit point; the set of integers, for example, lacks such a point. Every infinite bounded set of a Euclidean space, however, has at least one limit point.


Aleksandrov, P. S. Vvedenie v obshchuiu teoriiu mnozhestv i funktsil Moscow-Leningrad, 1948.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
A neutrosophic crisp point P [member of] [NCP.sub.n] in X is called a neutrosophic crisp limit point of B = < [B.sub.1], [B.sub.2], [B.sub.3] > iff every neutrosophic crisp open set containing P must contains at least one neutrosophic crisp point of B different from P.
It follows that [mathematical expression not reproducible], and hence both z and [M.sub.[lambda]]z are limit points of ([x.sub.n]).
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(d) Secant Poisson ratio (apparent incompressibility observed at limit point in compression).
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The following lemma investigates some properties of the dense limit points.
For the case of two explanatory features, the categorization was given in the case of regular boundaries by L+1 limit points gi for feature A and K+ 1 limit points [h.sub.j] for feature B.
Then, the limit points of the zeros of the polynomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are precisely the points of the line Re z = 1/2.
A set of points [LAMBDA} = [{[y.sub.n]}.sub.n[member of]Z] which has no limit points is called a set of sampling for the Hilbert space of [OMEGA]-bandlimited functions B([OMEGA]) if the norm squared of every bandlimited function is bounded above by the square sum of its samples taken on the points of [LAMBDA]:
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