View source for Locally closed subset

In [[topology]], a branch of mathematics, a subset <math>E</math> of a [[topological space]] <math>X</math> is said to be '''locally closed''' if any of the following equivalent conditions are satisfied:<ref name="Bourbaki">{{harvnb|Bourbaki|2007|loc=Ch. 1, § 3, no. 3.}}</ref><ref name="Explanation">{{harvnb|Pflaum|2001|loc=Explanation 1.1.2.}}</ref><ref>{{Cite journal|last1=Ganster |first1=M.|last2=Reilly|first2=I. L.|date=1989|title=Locally closed sets and LC -continuous functions|journal=International Journal of Mathematics and Mathematical Sciences|language=en|volume=12|issue=3|pages=417–424|doi=10.1155/S0161171289000505|issn=0161-1712|doi-access=free}}</ref>{{sfn|Engelking|1989|loc=Exercise 2.7.1}}
* <math>E</math> is the intersection of an open set and a closed set in <math>X.</math>
* For each point <math>x\in E,</math> there is a neighborhood <math>U</math> of <math>x</math> such that <math>E \cap U</math> is closed in <math>U.</math>
* <math>E</math> is open in its closure <math>\overline{E}.</math>
* The set <math>\overline{E}\setminus E</math> is closed in <math>X.</math>
* <math>E</math> is the difference of two closed sets in <math>X.</math>
* <math>E</math> is the difference of two open sets in <math>X.</math>

The second condition justifies the terminology ''locally closed'' and is Bourbaki's definition of locally closed.<ref name="Bourbaki" /> To see the second condition implies the third, use the facts that for subsets <math>A \subseteq B,</math> <math>A</math> is closed in <math>B</math> if and only if <math>A = \overline{A} \cap B</math> and that for a subset <math>E</math> and an open subset <math>U,</math> <math>\overline{E} \cap U = \overline{E \cap U} \cap U.</math>

== Examples ==
The interval <math>(0, 1] = (0, 2) \cap [0, 1]</math> is a locally closed subset of <math>\Reals.</math> For another example, consider the relative interior <math>D</math> of a closed disk in <math>\Reals^3.</math> It is locally closed since it is an intersection of the closed disk and an open ball.

On the other hand, <math>\{ (x,y)\in\Reals^2 \mid x\ne0 \} \cup \{(0,0)\}</math> is ''not'' a locally closed subset of <math>\Reals^2</math>.

Recall that, by definition, a submanifold <math>E</math> of an <math>n</math>-manifold <math>M</math> is a subset such that for each point <math>x</math> in <math>E,</math> there is a chart <math>\varphi : U \to \Reals^n</math> around it such that <math>\varphi(E \cap U) = \Reals^k \cap \varphi(U).</math> Hence, a submanifold is locally closed.<ref>{{cite journal |last1=Mather |first1=John |title=Notes on Topological Stability |journal=Bulletin of the American Mathematical Society |date=2012 |volume=49 |issue=4 |pages=475–506 |doi=10.1090/S0273-0979-2012-01383-6|doi-access=free }}section 1, p. 476</ref>

Here is an example in algebraic geometry. Let ''U'' be an open affine chart on a projective variety ''X'' (in the Zariski topology). Then each closed subvariety ''Y'' of ''U'' is locally closed in ''X''; namely, <math>Y = U \cap \overline{Y}</math> where <math>\overline{Y}</math> denotes the closure of ''Y'' in ''X''. (See also [[quasi-projective variety]] and [[quasi-affine variety]].)

==Properties==

Finite intersections and the pre-image under a continuous map of locally closed sets are locally closed.<ref name="Bourbaki" /> On the other hand, a union and a complement of locally closed subsets need not be locally closed.<ref>{{harvnb|Bourbaki|2007|loc=Ch. 1, § 3, Exercise 7.}}</ref> (This motivates the notion of a [[constructible set (topology)|constructible set]].)

Especially in [[stratification theory]], for a locally closed subset <math>E,</math> the complement <math>\overline{E} \setminus E</math> is called the '''boundary''' of <math>E</math> (not to be confused with [[topological boundary]]).<ref name="Explanation" /> If <math>E</math> is a closed submanifold-with-boundary of a manifold <math>M,</math> then the relative interior (that is, interior as a manifold) of <math>E</math> is locally closed in <math>M</math> and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset.<ref name="Explanation" />

A topological space is said to be '''{{visible anchor|submaximal|submaximal space}}''' if every subset is locally closed. See [[Glossary of topology#S]] for more of this notion.

==See also==

* {{annotated link|Countably generated space}}

==Notes==
{{reflist}}

==References==
*{{cite book |last1=Bourbaki |first1=Nicolas |title=Topologie générale. Chapitres 1 à 4 |date=2007 |publisher=Springer |location=Berlin |doi=10.1007/978-3-540-33982-3 |isbn=978-3-540-33982-3 |url=https://link.springer.com/book/10.1007/978-3-540-33982-3}}
* {{Bourbaki General Topology Part I Chapters 1-4}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{cite book|last=Engelking|first=Ryszard| author-link=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}
* {{cite book|last=Pflaum|first=Markus J.|title=Analytic and geometric study of stratified spaces|publisher=Springer|publication-place=Berlin|year=2001|series=Lecture Notes in Mathematics|volume=1768|isbn=3-540-42626-4|oclc=47892611}} <!--{{sfn|Pflaum|2001|p=}}-->

==External links==

* {{nlab|id=locally+closed+set|title=locally closed set}}

[[Category:General topology]]