Editing Open set (section)

== Special types of open sets ==

=== Clopen sets and non-open and/or non-closed sets ===

A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset {{em|and}} a closed subset. Such subsets are known as '''{{em|[[clopen set]]s}}'''. Explicitly,  a subset <math>S</math> of a topological space <math>(X, \tau)</math> is called {{em|clopen}} if both <math>S</math> and its complement <math>X \setminus S</math> are open subsets of <math>(X, \tau)</math>; or equivalently, if <math>S \in \tau</math> and <math>X \setminus S \in \tau.</math> 

In {{em|any}} topological space <math>(X, \tau),</math> the empty set <math>\varnothing</math> and the set <math>X</math> itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in {{em|every}} topological space. To see, it suffices to remark that, by definition of a topology, <math>X</math> and <math>\varnothing</math> are both open, and that they are also closed, since each is the complement of the other.

The open sets of the usual [[Euclidean topology]] of the [[real line]] <math>\R</math> are the empty set, the [[open interval]]s and every union of open intervals.

* The interval <math>I = (0, 1)</math> is open in <math>\R</math> by definition of the Euclidean topology. It is not closed since its complement in <math>\R</math> is <math> I^\complement = (-\infty, 0] \cup [1, \infty),</math> which is not open; indeed, an open interval contained in <math>I^\complement</math> cannot contain {{math|1}}, and it follows that <math>I^\complement</math> cannot be a union of open intervals. Hence, <math>I</math> is an example of a set that is open but not closed.
* By a similar argument, the interval <math>J = [0, 1]</math> is a closed subset but not an open subset.
* Finally, neither <math>K = [0, 1)</math> nor its complement <math>\R \setminus K = (-\infty, 0) \cup [1, \infty)</math> are open (because they cannot be written as a union of open intervals); this means that <math>K</math> is neither open nor closed.

If a topological space <math>X</math> is endowed with the [[discrete topology]] (so that by definition, every subset of <math>X</math> is open) then every subset of <math>X</math> is a clopen subset. 
For a more advanced example reminiscent of the discrete topology, suppose that <math>\mathcal{U}</math> is an [[ultrafilter]] on a non-empty set <math>X.</math> Then the union <math>\tau := \mathcal{U} \cup \{ \varnothing \}</math> is a topology on <math>X</math> with the property that {{em|every}} non-empty proper subset <math>S</math> of <math>X</math> is {{em|either}} an open subset or else a closed subset, but never both; that is, if <math>\varnothing \neq S \subsetneq X</math> (where <math>S \neq X</math>) then {{em|exactly one}} of the following two statements is true: either (1) <math>S \in \tau</math> or else, (2) <math>X \setminus S \in \tau.</math> Said differently, {{em|every}} subset is open or closed but the  {{em|only}} subsets that are both (i.e. that are clopen) are <math>\varnothing</math> and <math>X.</math>

=== Regular open sets{{anchor|Regular open set|Regular closed set}} ===

A subset <math>S</math> of a topological space <math>X</math> is called a '''{{em|[[regular open set]]}}''' if <math>\operatorname{Int} \left( \overline{S} \right) = S</math> or equivalently, if <math>\operatorname{Bd} \left( \overline{S} \right) = \operatorname{Bd} S</math>, where <math>\operatorname{Bd} S</math>, <math>\operatorname{Int} S</math>, and <math>\overline{S}</math> denote, respectively, the topological [[Boundary (topology)|boundary]], [[Interior (topology)|interior]], and [[Closure (topology)|closure]] of <math>S</math> in <math>X</math>.  A topological space for which there exists a [[Base (topology)|base]] consisting of regular open sets is called a '''{{em|[[semiregular space]]}}'''. 
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, where by definition a subset <math>S</math> of <math>X</math> is called a '''{{em|[[regular closed set]]}}''' if <math>\overline{\operatorname{Int} S} = S</math> or equivalently, if <math>\operatorname{Bd} \left( \operatorname{Int} S \right) = \operatorname{Bd} S.</math> 
Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,<ref group=note>One exception if the if <math>X</math> is endowed with the [[discrete topology]], in which case every subset of <math>X</math> is both a regular open subset and a regular closed subset of <math>X.</math></ref> the converses are {{em|not}} true.