Locally closed subset
In topology, a branch of mathematics, a subset [math]\displaystyle{ E }[/math] of a topological space [math]\displaystyle{ X }[/math] is said to be locally closed if any of the following equivalent conditions are satisfied:[1][2][3][4]
- [math]\displaystyle{ E }[/math] is the intersection of an open set and a closed set in [math]\displaystyle{ X. }[/math]
- For each point [math]\displaystyle{ x\in E, }[/math] there is a neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math] such that [math]\displaystyle{ E \cap U }[/math] is closed in [math]\displaystyle{ U. }[/math]
- [math]\displaystyle{ E }[/math] is an open subset of its closure [math]\displaystyle{ \overline{E}. }[/math]
- The set [math]\displaystyle{ \overline{E}\setminus E }[/math] is closed in [math]\displaystyle{ X. }[/math]
- [math]\displaystyle{ E }[/math] is the difference of two closed sets in [math]\displaystyle{ X. }[/math]
- [math]\displaystyle{ E }[/math] is the difference of two open sets in [math]\displaystyle{ X. }[/math]
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 [math]\displaystyle{ A \subseteq B, }[/math] [math]\displaystyle{ A }[/math] is closed in [math]\displaystyle{ B }[/math] if and only if [math]\displaystyle{ A = \overline{A} \cap B }[/math] and that for a subset [math]\displaystyle{ E }[/math] and an open subset [math]\displaystyle{ U, }[/math] [math]\displaystyle{ \overline{E} \cap U = \overline{E \cap U} \cap U. }[/math]
Examples
The interval [math]\displaystyle{ (0, 1] = (0, 2) \cap [0, 1] }[/math] is a locally closed subset of [math]\displaystyle{ \Reals. }[/math] For another example, consider the relative interior [math]\displaystyle{ D }[/math] of a closed disk in [math]\displaystyle{ \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]\displaystyle{ \{ (x,y)\in\Reals^2 \mid x\ne0 \} \cup \{(0,0)\} }[/math] is not a locally closed subset of [math]\displaystyle{ \Reals^2 }[/math].
Recall that, by definition, a submanifold [math]\displaystyle{ E }[/math] of an [math]\displaystyle{ n }[/math]-manifold [math]\displaystyle{ M }[/math] is a subset such that for each point [math]\displaystyle{ x }[/math] in [math]\displaystyle{ E, }[/math] there is a chart [math]\displaystyle{ \varphi : U \to \Reals^n }[/math] around it such that [math]\displaystyle{ \varphi(E \cap U) = \Reals^k \cap \varphi(U). }[/math] 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, [math]\displaystyle{ Y = U \cap \overline{Y} }[/math] where [math]\displaystyle{ \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.[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 [math]\displaystyle{ E, }[/math] the complement [math]\displaystyle{ \overline{E} \setminus E }[/math] is called the boundary of [math]\displaystyle{ E }[/math] (not to be confused with topological boundary).[2] If [math]\displaystyle{ E }[/math] is a closed submanifold-with-boundary of a manifold [math]\displaystyle{ M, }[/math] then the relative interior (that is, interior as a manifold) of [math]\displaystyle{ E }[/math] is locally closed in [math]\displaystyle{ M }[/math] 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 for more of this notion.
See also
Notes
- ↑ 1.0 1.1 1.2 Bourbaki 2007, Ch. 1, § 3, no. 3.
- ↑ 2.0 2.1 2.2 Pflaum 2001, Explanation 1.1.2.
- ↑ Ganster, M.; Reilly, I. L. (1989). "Locally closed sets and LC -continuous functions" (in en). International Journal of Mathematics and Mathematical Sciences 12 (3): 417–424. doi:10.1155/S0161171289000505. ISSN 0161-1712.
- ↑ Engelking 1989, Exercise 2.7.2.
- ↑ 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
- ↑ 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. https://link.springer.com/book/10.1007/978-3-540-33982-3.
- Bourbaki, Nicolas (1989). General Topology: Chapters 1–4. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129. https://doku.pub/documents/31425779-nicolas-bourbaki-general-topology-part-i1pdf-30j71z37920w.
- 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. 1768. Berlin: Springer. ISBN 3-540-42626-4. OCLC 47892611.
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