Locally closed subset

In topology, a branch of mathematics, a subset ${\displaystyle E}$ of a topological space ${\displaystyle X}$ is said to be locally closed if any of the following equivalent conditions are satisfied: ^{[1] [2] [3] [4]}

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

The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed. ^[1] To see the second condition implies the third, use the facts that for subsets ${\displaystyle A\subseteq B,}$${\displaystyle A}$ is closed in ${\displaystyle B}$ if and only if ${\displaystyle A={\overline {A}}\cap B}$ and that for a subset ${\displaystyle E}$ and an open subset ${\displaystyle U,}$${\displaystyle {\overline {E}}\cap U={\overline {E\cap U}}\cap U.}$

Examples

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

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

Recall that, by definition, a submanifold ${\displaystyle E}$ of an ${\displaystyle n}$-manifold ${\displaystyle M}$ is a subset such that for each point ${\displaystyle x}$ in ${\displaystyle E,}$ there is a chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$ around it such that ${\displaystyle \varphi (E\cap U)=\mathbb {R} ^{k}\cap \varphi (U).}$ Hence, a submanifold is locally closed. ^[5]

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, ${\displaystyle Y=U\cap {\overline {Y}}}$ where ${\displaystyle {\overline {Y}}}$ 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. ^[1] On the other hand, a union and a complement of locally closed subsets need not be locally closed. ^[6] (This motivates the notion of a constructible set.)

Especially in stratification theory, for a locally closed subset ${\displaystyle E,}$ the complement ${\displaystyle {\overline {E}}\setminus E}$ is called the boundary of ${\displaystyle E}$ (not to be confused with topological boundary). ^[2] If ${\displaystyle E}$ is a closed submanifold-with-boundary of a manifold ${\displaystyle M,}$ then the relative interior (that is, interior as a manifold) of ${\displaystyle E}$ is locally closed in ${\displaystyle M}$ and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset. ^[2]

A topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion.

See also

• Countably generated space  – topological space in which the topology is determined by its countable subsetsPages displaying wikidata descriptions as a fallback

Notes

1. Bourbaki 2007 , Ch. 1, § 3, no. 3.
2. Pflaum 2001 , Explanation 1.1.2.
3. Ganster, M.; Reilly, I. L. (1989). "Locally closed sets and LC -continuous functions". International Journal of Mathematics and Mathematical Sciences. 12 (3): 417–424. doi: 10.1155/S0161171289000505 . ISSN   0161-1712.
4. Engelking 1989, Exercise 2.7.2.
5. Mather, John (2012). "Notes on Topological Stability". Bulletin of the American Mathematical Society. 49 (4): 475–506. doi: 10.1090/S0273-0979-2012-01383-6 .section 1, p. 476
6. Bourbaki 2007 , Ch. 1, § 3, Exercise 7.

References

• Bourbaki, Nicolas (2007). Topologie générale. Chapitres 1 à 4. Berlin: Springer. doi:10.1007/978-3-540-33982-3. ISBN   978-3-540-33982-3.
• Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [ Topologie Générale ]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN   978-3-540-64241-1. OCLC   18588129.
• Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN   3-88538-006-4.
• Pflaum, Markus J. (2001). Analytic and geometric study of stratified spaces. Lecture Notes in Mathematics. Vol. 1768. Berlin: Springer. ISBN   3-540-42626-4. OCLC   47892611.

External links

• locally closed set at the nLab