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Difference between revisions of "Interior of a set"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518501.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518502.png" />''
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{{MSC|54A}}
  
The set of interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518503.png" /> (cf. [[Interior point of a set|Interior point of a set]]). It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518504.png" />. Invariably, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518506.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518507.png" />. The interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518508.png" /> is also equal to the union of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518509.png" /> that are open in the entire space. The interior of a set is sometimes known as the open kernel.
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''of a set $A$ in a [[topological space]] $X$''
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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 $\langle A \rangle$.
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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 $X \setminus \langle B \rangle = [ A ]$.
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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.
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The terminology  "kernel" is seldom used in this context in the modern English mathematical literature.
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==References==
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> Franz, Wolfgang. ''General topology'' (Harrap, 1967).</TD></TR>
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<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>
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</table>

Latest revision as of 16:55, 25 November 2023

2020 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 of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior_of_a_set&oldid=14280
This article was adapted from an original article by S.M. Sirota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article