Locally closed subset

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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:[1][2][3][4]

  • 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.[1] 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

Examples[edit]

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.[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, where denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)

Properties[edit]

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 the complement is called the boundary of (not to be confused with topological boundary).[2] 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.[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[edit]

  • Countably generated space – topological space in which the topology is determined by its countable subsets

Notes[edit]

  1. ^ a b c Bourbaki 2007, Ch. 1, § 3, no. 3.
  2. ^ a b c 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[edit]

External links[edit]