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Difference between revisions of "Boundary"

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The boundary of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172501.png" /> of a given topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172502.png" /> is the set of points such that every neighbourhood of any point of it contains both points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172503.png" /> and points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172504.png" />. The accepted notations include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172507.png" />.
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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.
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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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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=39404
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article