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Interior of a set

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2020 Mathematics Subject Classification: Primary: 54A [MSN][ZBL]

of a set 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=54692
This article was adapted from an original article by S.M. Sirota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article