Adherent point - Citizendia

In mathematics, an adherent point (also called a closure point or point of closure) is a slight generalization of the idea of a limit point. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"

Let $X$ be a topological space and $A\subset X$ be a subset. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. A point $x\in X$ is an adherent point for $A$ if every open set containing $x$ contains at least one point of $A$ . In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in A point $x$ is an adherent point for $A$ if and only if $x$ is in the closure of $A$ . In Mathematics, the closure of a set S consists of all points which are intuitively "close to S "

This definition is more general than that of a limit point, in that for a limit point it is required that every open set containing $x$ contains at least one point of $A$ different from $x$ . In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" Thus every limit point is an adherent point, but the converse fails. An adherent point which is not a limit point is an isolated point. In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of

References

L. A. Steen, J. A. Seebach, Jr. , Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc. .
This article incorporates material from Adherent point on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |