Editing Open set (section)

== Generalizations of open sets ==
{{See also|Almost open map|Glossary of topology}}

Throughout, <math>(X, \tau)</math> will be a topological space. 

A subset <math>A \subseteq X</math> of a topological space <math>X</math> is called:

<ul>
<li>'''{{em|α-open}}''' if <math>A ~\subseteq~ \operatorname{int}_X \left( \operatorname{cl}_X \left( \operatorname{int}_X A \right) \right)</math>, and the complement of such a set is called '''{{em|α-closed}}'''.{{sfn|Hart|2004|p=9}}</li>
<li>'''{{em|preopen}}''', '''{{em|nearly open}}''', or '''{{em|locally [[Dense subset|dense]]}}''' if it satisfies any of the following equivalent conditions:
<ol>
<li><math>A ~\subseteq~ \operatorname{int}_X \left( \operatorname{cl}_X A \right).</math>{{sfn|Hart|2004|pp=8–9}}</li>
<li>There exists subsets <math>D, U \subseteq X</math> such that <math>U</math> is open in <math>X,</math> <math>D</math> is a [[dense subset]] of <math>X,</math> and <math>A = U \cap D.</math>{{sfn|Hart|2004|pp=8–9}}</li>
<li>There exists an open (in <math>X</math>) subset <math>U \subseteq X</math> such that <math>A</math> is a dense subset of <math>U.</math>{{sfn|Hart|2004|pp=8–9}}</li>
</ol>
The complement of a preopen set is called '''{{em|preclosed}}'''. 
</li>
<li>'''{{em|b-open}}''' if <math>A ~\subseteq~ \operatorname{int}_X \left( \operatorname{cl}_X A \right) ~\cup~ \operatorname{cl}_X \left( \operatorname{int}_X A \right)</math>. The complement of a b-open set is called '''{{em|b-closed}}'''.{{sfn|Hart|2004|p=9}}</li>
<li>'''{{em|β-open}}''' or '''{{em|semi-preopen}}''' if it satisfies any of the following equivalent conditions:
<ol>
<li><math>A ~\subseteq~ \operatorname{cl}_X \left( \operatorname{int}_X \left( \operatorname{cl}_X A \right) \right)</math>{{sfn|Hart|2004|p=9}}</li>
<li><math> \operatorname{cl}_X A</math> is a regular closed subset of <math>X.</math>{{sfn|Hart|2004|pp=8–9}}</li>
<li>There exists a preopen subset <math>U</math> of <math>X</math> such that <math>U \subseteq A \subseteq \operatorname{cl}_X U.</math>{{sfn|Hart|2004|pp=8–9}}</li>
</ol>
The complement of a β-open set is called '''{{em|β-closed}}'''. 
</li>
<li>'''{{em|[[sequentially open]]}}''' if it satisfies any of the following equivalent conditions:
<ol>
<li>Whenever a sequence in <math>X</math> converges to some point of <math>A,</math> then that sequence is eventually in <math>A.</math> Explicitly, this means that if <math>x_{\bull} = \left( x_i \right)_{i=1}^{\infty}</math> is a sequence in <math>X</math> and if there exists some <math>a \in A</math> is such that <math>x_{\bull} \to x</math> in <math>(X, \tau),</math> then <math>x_{\bull}</math> is eventually in <math>A</math> (that is, there exists some integer <math>i</math> such that if <math>j \geq i,</math> then <math>x_j \in A</math>).</li>
<li><math>A</math> is equal to its '''{{em|sequential interior}}''' in <math>X,</math> which by definition is the set
:<math>\begin{alignat}{4}
\operatorname{SeqInt}_X A
:&= \{ a \in A ~:~ \text{ whenever a sequence in } X \text{ converges to } a \text{ in } (X, \tau), \text{ then that sequence is eventually in } A \} \\
&= \{ a \in A ~:~ \text{ there does NOT exist a sequence in } X \setminus A \text{ that converges in } (X, \tau) \text{ to a point in } A \} \\
\end{alignat}
</math>
</li>
</ol>
The complement of a sequentially open set is called '''{{em|sequentially  closed}}'''. A subset <math>S \subseteq X</math> is sequentially closed in <math>X</math> if and only if <math>S</math> is equal to its '''{{em|sequential closure}}''', which by definition is the set <math>\operatorname{SeqCl}_X S</math> consisting of all <math>x \in X</math> for which there exists a sequence in <math>S</math> that converges to <math>x</math> (in <math>X</math>). 
</li>
<li>'''{{em|[[Almost open set|almost open]]}}''' and is said to have '''{{em|the Baire property}}''' if there exists an open subset <math>U \subseteq X</math> such that <math>A \bigtriangleup U</math> is a [[Meager set|meager subset]], where <math>\bigtriangleup</math> denotes the [[symmetric difference]].<ref name="oxtoby">{{citation|title=Measure and Category|volume=2|series=Graduate Texts in Mathematics|first=John C.|last=Oxtoby|edition=2nd|publisher=Springer-Verlag|year=1980|isbn=978-0-387-90508-2|contribution=4. The Property of Baire|pages=19–21|url=https://books.google.com/books?id=wUDjoT5xIFAC&pg=PA19}}.</ref>
* The subset <math>A \subseteq X</math> is said to have '''the Baire property in the restricted sense''' if for every subset <math>E</math> of <math>X</math> the intersection <math>A\cap E</math> has the Baire property relative to <math>E</math>.<ref>{{citation|last=Kuratowski|first=Kazimierz|authorlink=Kazimierz Kuratowski|title= Topology. Vol. 1|publisher=Academic Press and Polish Scientific Publishers|year=1966}}.</ref></li>
<li>'''{{em|semi-open}}''' if <math>A ~\subseteq~ \operatorname{cl}_X \left( \operatorname{int}_X A \right)</math> or, equivalently, <math>\operatorname{cl}_X A = \operatorname{cl}_X \left( \operatorname{int}_X A \right)</math>. The complement in <math>X</math> of a semi-open set is called a '''{{em|semi-closed}} set'''.{{sfn|Hart|2004|p=8}} 
* The '''{{em|semi-closure}}''' (in <math>X</math>) of a subset <math>A \subseteq X,</math> denoted by <math>\operatorname{sCl}_X A,</math> is the intersection of all semi-closed subsets of <math>X</math> that contain <math>A</math> as a subset.{{sfn|Hart|2004|p=8}}</li>
<li>'''{{em|semi-θ-open}}''' if for each <math>x \in A</math> there exists some semiopen subset <math>U</math> of <math>X</math> such that <math>x \in U \subseteq \operatorname{sCl}_X U \subseteq A.</math>{{sfn|Hart|2004|p=8}}</li>
<li>'''{{em|θ-open}}''' (resp. '''{{em|δ-open}}''') if its complement in <math>X</math> is a θ-closed (resp. {{em|δ-closed}}) set, where by definition, a subset of <math>X</math> is called '''{{em|θ-closed}}''' (resp. '''{{em|δ-closed}}''') if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point <math>x \in X</math> is called a '''{{em|θ-cluster point}}''' (resp. a '''{{em|δ-cluster point}}''') of a subset <math>B \subseteq X</math> if for every open neighborhood <math>U</math> of <math>x</math> in <math>X,</math> the intersection <math>B \cap \operatorname{cl}_X U</math> is not empty (resp. <math>B \cap \operatorname{int}_X\left( \operatorname{cl}_X U \right)</math> is not empty).{{sfn|Hart|2004|p=8}}</li>
</ul>

Using the fact that 
:<math>A ~\subseteq~ \operatorname{cl}_X A ~\subseteq~ \operatorname{cl}_X B</math> {{spaces|4}}and{{spaces|4}} <math>\operatorname{int}_X A ~\subseteq~ \operatorname{int}_X B ~\subseteq~ B</math>

whenever two subsets <math>A, B \subseteq X</math> satisfy <math>A \subseteq B,</math> the following may be deduced:

* Every α-open subset is semi-open, semi-preopen, preopen, and b-open. 
* Every b-open set is semi-preopen (i.e. β-open). 
* Every preopen set is b-open and semi-preopen. 
* Every semi-open set is b-open and semi-preopen.

Moreover, a subset is a regular open set if and only if it is preopen and semi-closed.{{sfn|Hart|2004|pp=8–9}} The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set.{{sfn|Hart|2004|pp=8–9}}  Preopen sets need not be semi-open and semi-open sets need not be preopen.{{sfn|Hart|2004|pp=8–9}} 

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).{{sfn|Hart|2004|pp=8-9}} However, finite intersections of preopen sets need not be preopen.{{sfn|Hart|2004|p=8}} The set of all α-open subsets of a space <math>(X, \tau)</math> forms a topology on <math>X</math> that is [[Comparison of topologies|finer]] than <math>\tau.</math>{{sfn|Hart|2004|p=9}} 

A topological space <math>X</math> is [[Hausdorff space|Hausdorff]] if and only if every [[Compact space|compact subspace]] of <math>X</math> is θ-closed.{{sfn|Hart|2004|p=8}} 
A space <math>X</math> is [[totally disconnected]] if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the '''{{em|[[Closure (topology)|closure]]}}''' of every preopen subset is open.{{sfn|Hart|2004|p=9}}