Locally closed subset explained

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:^[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\ne 0 \} \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.^[2]

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. (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 if every subset is locally closed. See Glossary of topology#S for more of this notion.

References

• Book: Bourbaki . Nicolas . Topologie générale. Chapitres 1 à 4 . 2007 . Springer . Berlin . 10.1007/978-3-540-33982-3 . 978-3-540-33982-3 .
  • Book: Engelking, Ryszard. Ryszard Engelking. General Topology. Heldermann Verlag, Berlin. 1989. 3-88538-006-4.
• Book: Pflaum, Markus J.. Analytic and geometric study of stratified spaces. Springer. Berlin. 2001. Lecture Notes in Mathematics. 1768. 3-540-42626-4. 47892611.

Notes and References

1. Ganster . M.. Reilly. I. L.. 1989. Locally closed sets and LC -continuous functions. International Journal of Mathematics and Mathematical Sciences. en. 12. 3. 417–424. 10.1155/S0161171289000505. 0161-1712. free.
2. Mather . John . Notes on Topological Stability . Bulletin of the American Mathematical Society . 2012 . 49 . 4 . 475–506 . 10.1090/S0273-0979-2012-01383-6. free . section 1, p. 476