Locally closed subset
In topology, a branch of mathematics, a subset of a topological space is said to be locally closed if any of the following equivalent conditions are satisfied:
- is the intersection of an open set and a closed set in
- For each point there is a neighborhood of such that is closed in
- is an open subset of its closure
- The set is closed in
- is the difference of two closed sets in
- is the difference of two open sets in
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 is closed in if and only if and that for a subset and an open subset
The interval is a locally closed subset of For another example, consider the relative interior of a closed disk in It is locally closed since it is an intersection of the closed disk and an open ball.
Recall that, by definition, a submanifold of an -manifold is a subset such that for each point in there is a chart around it such that Hence, a submanifold is locally closed.
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, where denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)
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. (This motivates the notion of a constructible set.)
Especially in stratification theory, for a locally closed subset the complement is called the boundary of (not to be confused with topological boundary). If is a closed submanifold-with-boundary of a manifold then the relative interior (that is, interior as a manifold) of is locally closed in 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 submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion.
- Countably generated space – topological space in which the topology is determined by its countable subsets
- ^ a b c Bourbaki 2007, Ch. 1, § 3, no. 3. harvnb error: no target: CITEREFBourbaki2007 (help)
- ^ a b c Pflaum 2001, Explanation 1.1.2.
- ^ 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.
- ^ 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. harvnb error: no target: CITEREFBourbaki2007 (help)
- Bourbaki, Topologie générale, 2007.
- Bourbaki, Nicolas (1989) . 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.
- locally closed set at the nLab