Editing Boundary (topology) (section)

=== Characterizations and general examples ===

A set and its complement have the same boundary:
<math display="block">\partial_X S = \partial_X (X \setminus S).</math>

A set <math>U</math> is a [[Dense subset|dense]] [[Open set|open]] subset of <math>X</math> if and only if <math>\partial_X U = X \setminus U.</math> 

The interior of the boundary of a closed set is empty.<ref group="proof">Let <math>S</math> be a closed subset of <math>X</math> so that <math>\overline{S} = S</math> and thus also <math>\partial_X S := \overline{S} \setminus \operatorname{int}_X S = S \setminus \operatorname{int}_X S.</math> If <math>U</math> is an open subset of <math>X</math> such that <math>U \subseteq \partial_X S</math> then  <math>U \subseteq S</math> (because <math>\partial_X S \subseteq S</math>) so that <math>U \subseteq \operatorname{int}_X S</math> (because [[Interior (topology)|by definition]], <math>\operatorname{int}_X S</math> is the largest open subset of <math>X</math> contained in <math>S</math>). But <math>U \subseteq \partial_X S = S \setminus \operatorname{int}_X S</math> implies that <math>U \cap \operatorname{int}_X S = \varnothing.</math> Thus <math>U</math> is simultaneously a subset of <math>\operatorname{int}_X S</math> and disjoint from <math>\operatorname{int}_X S,</math> which is only possible if <math>U = \varnothing.</math> [[Q.E.D.]]</ref> 
Consequently, the interior of the boundary of the closure of a set is empty. 
The interior of the boundary of an open set is also empty.<ref group="proof">Let <math>S</math> be an open subset of <math>X</math> so that <math>\partial_X S := \overline{S} \setminus \operatorname{int}_X S = \overline{S} \setminus S.</math> Let <math>U := \operatorname{int}_X \left(\partial_X S\right)</math> so that <math>U = \operatorname{int}_X \left(\partial_X S\right) \subseteq \partial_X S = \overline{S} \setminus S,</math> which implies that <math>U \cap S = \varnothing.</math> If <math>U \neq \varnothing</math> then pick <math>u \in U,</math> so that <math>u \in U \subseteq \partial_X S \subseteq \overline{S}.</math> Because <math>U</math> is an open neighborhood of <math>u</math> in <math>X</math> and <math>u \in \overline{S},</math> the definition of the [[Closure (topology)|topological closure]] <math>\overline{S}</math> implies that <math>U \cap S \neq \varnothing,</math> which is a contradiction. <math>\blacksquare</math> Alternatively, if <math>S</math> is open in <math>X</math> then <math>X \setminus S</math> is closed in <math>X,</math> so that by using the general formula <math>\partial_X S = \partial_X (X \setminus S)</math> and the fact that the interior of the boundary of a closed set (such as <math>X \setminus S</math>) is empty, it follows that <math>\operatorname{int}_X \partial_X S = \operatorname{int}_X \partial_X (X \setminus S) = \varnothing.</math> <math>\blacksquare</math></ref> 
Consequently, the interior of the boundary of the interior of a set is empty. 
In particular, if <math>S \subseteq X</math> is a closed or open subset of <math>X</math> then there does not exist any nonempty subset <math>U \subseteq \partial_X S</math> such that <math>U</math> is open in <math>X.</math> 
This fact is important for the definition and use of [[Nowhere dense set|nowhere dense subsets]], [[Meager set|meager subsets]], and [[Baire space]]s. 

A set is the boundary of some open set if and only if it is closed and [[Nowhere dense set|nowhere dense]]. 
The boundary of a set is empty if and only if the set is both closed and open (that is, a [[clopen set]]).