View source for Interior of a set

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{{MSC|54A}}

''of a set $A$ in a [[topological space]] $X$''

The interior, or (open) kernel, of $A$ is the set of all [[Interior point of a set|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 a set|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==
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> Franz, Wolfgang. ''General topology'' (Harrap, 1967).</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> John L. Kelley, ''General Topology'', Graduate Texts in Mathematics '''27''', Springer (1975) {{ISBN|0-387-90125-6}}</TD></TR>
</table>