Locally closed subset

Short description: Intersection of an open set and a closed set

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:

• $E$ 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 $\bar{E}.$
• The set $\bar{E}\setminus E$ 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 $A=\bar{A}\cap B$ and that for a subset $E$ and an open subset $U,$ $\bar{E}\cap U=\overline{E\cap U}\cap U.$

Examples

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

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

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 $\phi :U\to {\mathbb{ℝ}}^n$ around it such that $\phi (E\cap U)={\mathbb{ℝ}}^k\cap \phi (U).$ 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, $Y=U\cap \bar{Y}$ where $\bar{Y}$ 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 $\bar{E}\setminus E$ 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 submaximal if every subset is locally closed. See Glossary of topology for more of this notion.

See also

• Countably generated space

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. 

External links

• locally closed set in nLab