# Boundary

2020 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.

#### 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=54720