Difference between revisions of "Boundary"
From Encyclopedia of Mathematics
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| − | The boundary of a subspace | + | The boundary of a subspace $A$ of a given [[topological space]] $X$ is the set of points of $X$ such that every [[neighbourhood]] of any point of it contains both points from $A$ and points from the complement $X\setminus A$. Equivalently, the points which are in the [[Interior of a set|interior]] neither of $A$ nor of $X \setminus A$; the set of points in the [[Closure of a set|closure]] of $A$ that are not in the interior of $A$. |
| − | A synonym for the border of a [[manifold]], such as the border of a simplex. | + | A subset $A$ is closed if it contains its boundary, and open if it is disjoint from its boundary. |
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| + | The accepted notations include $\partial A$, $b(A)$, $\mathrm{Fr}(A)$, $\mathrm{Fr}_X(A)$. | ||
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| + | Also: a synonym for the border of a [[manifold]], such as the border of a simplex. | ||
====References==== | ====References==== | ||
* J.L. Kelley, "General topology", Graduate Texts in Mathematics '''27''' Springer (1975) ISBN 0-387-90125-6 {{ZBL|0306.54002}} | * J.L. Kelley, "General topology", Graduate Texts in Mathematics '''27''' Springer (1975) ISBN 0-387-90125-6 {{ZBL|0306.54002}} | ||
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| + | {{TEX|done}} | ||
Revision as of 18:13, 10 October 2016
The boundary of a subspace $A$ of a given topological space $X$ is the set of points of $X$ such that every neighbourhood of any point of it contains both points from $A$ and points from the complement $X\setminus A$. Equivalently, the points which are in the interior neither of $A$ nor of $X \setminus A$; the set of points in the closure of $A$ that are not in the interior of $A$.
A subset $A$ is closed if it contains its boundary, and open if it is disjoint from its boundary.
The accepted notations include $\partial A$, $b(A)$, $\mathrm{Fr}(A)$, $\mathrm{Fr}_X(A)$.
Also: a synonym for the border of a manifold, such as the border of a simplex.
References
- J.L. Kelley, "General topology", Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
How to Cite This Entry:
Boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary&oldid=39406
Boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary&oldid=39406
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article