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2010 Mathematics Subject Classification: Primary: 54A [MSN][ZBL]

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.


  • J.L. Kelley, "General topology", Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
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This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article