# Interior of a set

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

*of a set $A$ in a topological space $X$*

The interior, or (open) kernel, of $A$ is the set of all interior points of $A$: the union of all open sets of $X$ which are subsets of $A$; a point $x \in A$ is interior if there is a neighbourhood $N_x$ contained in $A$ and containing $x$. The interior may be denoted $A^\circ$, $\mathrm{Int} A$ or $\langle A \rangle$.

The interior of $A$ is the complement in $A$ of the boundary of $A$. If $A$ and $B$ are mutually complementary sets in a topological space $X$, that is, if $B = X \setminus A$, then the interior of $A$ is the complement of the closure of $B$: $X \setminus [A] = \langle B \rangle$ and $X \setminus \langle B \rangle = [ A ]$.

The interior of a closed set in a topological space $X$ is a regular open or canonical set. Spaces in which the open canonical sets form a base for the topology are called semi-regular. Every regular space is semi-regular.

The terminology "kernel" is seldom used in this context in the modern English mathematical literature.

## References

[1] | Franz, Wolfgang. General topology (Harrap, 1967). |

[2] | John L. Kelley, General Topology, Graduate Texts in Mathematics 27, Springer (1975) ISBN 0-387-90125-6 |

**How to Cite This Entry:**

Interior.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Interior&oldid=36993