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The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular closed set.
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is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton
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723:{\displaystyle {\overline {\operatorname {Int} \{x\}}}={\overline {\varnothing }}=\varnothing \neq \{x\}.}
1141: – topological space in which a point and a closed set are, if disjoint, separable by neighborhoods
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1206:. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.
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itself) is simultaneously a regular open subset and regular closed subset.
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if it is equal to the closure of its interior; expressed symbolically, if
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1025:{\displaystyle U\vee V=\operatorname {Int} ({\overline {U\cup V}}),}
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but not a regular closed set because its interior is the empty set
551:{\displaystyle \operatorname {Int} ({\overline {S}})=(0,2)\neq S.}
1189:
Willard, "3D, Regularly open and regularly closed sets", p. 29
396:{\displaystyle \partial (\operatorname {Int} S)=\partial S.}
1155: – Axioms in topology defining notions of "separation"
1135: – List of concrete topologies and topological spaces
1117:{\displaystyle \neg U=\operatorname {Int} (X\setminus U).}
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is a regular closed set. Every regular open set is an
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is a regular open set if and only if its complement in
109:{\displaystyle \operatorname {Int} ({\overline {S}})=S}
157:{\displaystyle \partial ({\overline {S}})=\partial S,}
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349:{\displaystyle {\overline {\operatorname {Int} S}}=S}
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186:{\displaystyle \operatorname {Int} S,}
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1067:{\displaystyle U\land V=U\cap V}
212:{\displaystyle {\overline {S}}}
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737:A subset of
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783:closed set
733:Properties
1103:∖
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271:A subset
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1259:Category
1127:See also
962:forms a
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655:so that
406:Examples
244:boundary
65:interior
69:closure
67:of its
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558:Every
164:where
1160:Notes
788:Each
36:of a
1244:OCLC
1234:ISBN
1208:ISBN
1034:meet
1032:the
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832:and
219:and
1091:Int
1036:is
990:Int
792:of
666:Int
562:in
500:Int
410:If
370:Int
326:Int
291:of
246:of
172:Int
79:Int
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1180:^
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