Regular open set: Difference between revisions

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A subset of <math>X</math> is a regular open set if and only if its complement in <math>X</math> is a regular closed set.<ref name="willard-regopen"/>  Every regular open set is an [[open set]] and every regular closed set is a [[closed set]].
A subset of <math>X</math> is a regular open set if and only if its complement in <math>X</math> is a regular closed set.<ref name="willard-regopen"/>  Every regular open set is an [[open set]] and every regular closed set is a [[closed set]].
A subset <math>G</math> in a topological space <math>X</math> is a regular open set if and only if  <math>G=\operatorname{Int}(\overline{A})</math> for some <math>A\subset X</math><ref name="willard-regopen">Willard, "3D, Regularly open and regularly closed sets", p. 29</ref>. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to
<math>\begin{aligned}
\operatorname{Int}(\overline{A})\subset \overline{\operatorname{Int}(\overline{A})} \quad \Longrightarrow \quad \operatorname{Int}(\overline{A})\subset \operatorname{Int}\Big( \overline{\operatorname{Int}(\overline{A})}\Big)
\end{aligned}</math>
<math>\begin{aligned}
\operatorname{Int}(\overline{A})\subset \overline{A} \quad \Longrightarrow \quad \overline{\operatorname{Int}(\overline{A})}\subset \overline{A}\quad\Longrightarrow\quad
\operatorname{Int}\Big( \overline{\operatorname{Int}(\overline{A})}\Big)\subset \operatorname{Int}(\overline{A})
\end{aligned}</math>


Each [[clopen subset]] of <math>X</math> (which includes <math>\varnothing</math> and <math>X</math> itself) is simultaneously a regular open subset and regular closed subset.
Each [[clopen subset]] of <math>X</math> (which includes <math>\varnothing</math> and <math>X</math> itself) is simultaneously a regular open subset and regular closed subset.


The interior of a closed subset of <math>X</math> is a regular open subset of <math>X</math> and likewise, the closure of an open subset of <math>X</math> is a regular closed subset of <math>X.</math><ref name="willard-regopen">Willard, "3D, Regularly open and regularly closed sets", p. 29</ref>  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.<ref name="willard-regopen"/>
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.<ref name="willard-regopen"/>


The collection of all regular open sets in <math>X</math> forms a [[complete Boolean algebra]]; the [[Join and meet|join]] operation is given by <math>U \vee V = \operatorname{Int}(\overline{U \cup V}),</math> the [[Join and meet|meet]] is <math>U \and V = U \cap V</math> and the complement is <math>\neg U = \operatorname{Int}(X \setminus U).</math>
The collection of all regular open sets in <math>X</math> forms a [[complete Boolean algebra]]; the [[Join and meet|join]] operation is given by <math>U \vee V = \operatorname{Int}(\overline{U \cup V}),</math> the [[Join and meet|meet]] is <math>U \and V = U \cap V</math> and the complement is <math>\neg U = \operatorname{Int}(X \setminus U).</math>
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==See also==
==See also==


* {{annotated link|List of topologies}}
* {{annotated link|Regular space}}
* {{annotated link|Regular space}}
* {{annotated link|Semiregular space}}
* {{annotated link|Semiregular space}}

Revision as of 01:50, 4 December 2025

A subset S of a topological space X is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if Int⁡(S‾)=S or, equivalently, if ∂(S‾)=∂S, where Int⁡S, S‾ and ∂S denote, respectively, the interior, closure and boundary of S.[1]

A subset S of X is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if Int⁡S‾=S or, equivalently, if ∂(Int⁡S)=∂S.[1]

Examples

If ℝ has its usual Euclidean topology then the open set S=(0,1)∪(1,2) is not a regular open set, since Int⁡(S‾)=(0,2)≠S. Every open interval in ℝ 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 {x} is a closed subset of ℝ but not a regular closed set because its interior is the empty set ∅, so that Int⁡{x}‾=∅‾=∅≠{x}.

Properties

A subset of X is a regular open set if and only if its complement in X is a regular closed set.[2] Every regular open set is an open set and every regular closed set is a closed set.

A subset G in a topological space X is a regular open set if and only if G=Int⁡(A‾) for some A⊂X[2]. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to

Int⁡(A‾)⊂Int⁡(A‾)‾⟹Int⁡(A‾)⊂Int⁡(Int⁡(A‾)‾)

Int⁡(A‾)⊂A‾⟹Int⁡(A‾)‾⊂A‾⟹Int⁡(Int⁡(A‾)‾)⊂Int⁡(A‾)

Each clopen subset of X (which includes ∅ and X itself) is simultaneously a regular open subset and regular closed subset.

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.[2]

The collection of all regular open sets in X forms a complete Boolean algebra; the join operation is given by U∨V=Int⁡(U∪V‾), the meet is U∧V=U∩V and the complement is ¬U=Int⁡(X∖U).

See also

Notes

<templatestyles src="Reflist/styles.css" />

  1. ↑ a b Steen & Seebach, p. 6
  2. ↑ a b c Willard, "3D, Regularly open and regularly closed sets", p. 29

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References

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