Difference between revisions of "Interior of a set"
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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 $\langle A \rangle$. | 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 $A$ 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 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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==References== | ==References== | ||
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| − | <TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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"> | + | <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> | </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 |
Interior of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior_of_a_set&oldid=37000