locally closed subset

In topology, a branch of mathematics, a subset $E$ of a topological space $X$ is said to be locally closed if any of the following equivalent conditions are satisfied:{{harvnb|Bourbaki|2007|loc=Ch. 1, § 3, no. 3.}}{{harvnb|Pflaum|2001|loc=Explanation 1.1.2.}}{{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}}{{sfn|Engelking|1989|loc=Exercise 2.7.1}}

$E$ is the intersection of an open set and a closed set in $X.$
For each point $x\in E,$ there is a neighborhood $U$ of $x$ such that $E \cap U$ is closed in $U.$
$E$ is open in its closure $\overline{E}.$
The set $\overline{E}\setminus E$ is closed in $X.$
$E$ is the difference of two closed sets in $X.$
$E$ is the difference of two open sets in $X.$

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

Examples

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

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

Recall that, by definition, a submanifold $E$ of an $n$ -manifold $M$ is a subset such that for each point $x$ in $E,$ there is a chart $\varphi : U \to \Reals^n$ around it such that $\varphi(E \cap U) = \Reals^k \cap \varphi(U).$ Hence, a submanifold is locally closed.{{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

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, $Y = U \cap \overline{Y}$ where $\overline{Y}$ 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. On the other hand, a union and a complement of locally closed subsets need not be locally closed.{{harvnb|Bourbaki|2007|loc=Ch. 1, § 3, Exercise 7.}} (This motivates the notion of a constructible set.)

Especially in stratification theory, for a locally closed subset $E,$ the complement $\overline{E} \setminus E$ is called the boundary of $E$ (not to be confused with topological boundary). If $E$ is a closed submanifold-with-boundary of a manifold $M,$ then the relative interior (that is, interior as a manifold) of $E$ is locally closed in $M$ and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset.

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.

Notes

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}}
{{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}}

External links

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

Category:General topology