Boundary

This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.

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 Template:ZBL