Editing Interior (topology) (section)

==Interior operator==
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The '''interior operator''' <math>\operatorname{int}_X</math> is dual to the [[Closure (topology)|closure]] operator, which is denoted by <math>\operatorname{cl}_X</math> or by an overline <sup>—</sup>, in the sense that
<math display="block">\operatorname{int}_X S = X \setminus \overline{(X \setminus S)}</math>
and also
<math display="block">\overline{S} = X \setminus \operatorname{int}_X (X \setminus S),</math>
where <math>X</math> is the [[topological space]] containing <math>S,</math> and the backslash <math>\,\setminus\,</math> denotes [[Complement (set theory)|set-theoretic difference]].
Therefore, the abstract theory of closure operators and the [[Kuratowski closure axioms]] can be readily translated into the language of interior operators, by replacing sets with their complements in <math>X.</math>

In general, the interior operator does not commute with unions. However, in a [[complete metric space]] the following result does hold:

{{Math theorem|name=Theorem<ref name="Zalinescu 2002 p. 33">{{cite book|last=Zalinescu|first=C|title=Convex analysis in general vector spaces|publisher=World Scientific|publication-place=River Edge, N.J. London|year=2002|isbn=981-238-067-1| oclc=285163112| page=33}}</ref> |note=C. Ursescu|math_statement=
Let <math>S_1, S_2, \ldots</math> be a sequence of subsets of a [[complete metric space]] <math>X.</math>
*If each <math>S_i</math> is closed in <math>X</math> then <math display="block">
{\operatorname{cl}_X} \biggl( \bigcup_{i \in \N} \operatorname{int}_X S_i \biggr)
= {\operatorname{cl}_X \operatorname{int}_X} \biggl( \bigcup_{i \in \N} S_i \biggr).
</math>
*If each <math>S_i</math> is open in <math>X</math> then <math display="block">
{\operatorname{int}_X} \biggl( \bigcap_{i \in \N} \operatorname{cl}_X S_i \biggr)
= {\operatorname{int}_X \operatorname{cl}_X} \biggl( \bigcap_{i \in \N} S_i \biggr).
</math>
}}

The result above implies that every complete metric space is a [[Baire space]].