Locally closed subset

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

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

Recall that, by definition, a submanifold ${\displaystyle E}$ of an ${\displaystyle n}$-manifold ${\displaystyle M}$ is a subset such that for each point ${\displaystyle x}$ in ${\displaystyle E,}$ there is a chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$ around it such that ${\displaystyle \varphi (E\cap U)=\mathbb {R} ^{k}\cap \varphi (U).}$ 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, ${\displaystyle Y=U\cap {\overline {Y}}}$ where ${\displaystyle {\overline {Y}}}$ 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.[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 ${\displaystyle E,}$ the complement ${\displaystyle {\overline {E}}\setminus E}$ is called the boundary of ${\displaystyle E}$ (not to be confused with topological boundary).[2] If ${\displaystyle E}$ is a closed submanifold-with-boundary of a manifold ${\displaystyle M,}$ then the relative interior (that is, interior as a manifold) of ${\displaystyle E}$ is locally closed in ${\displaystyle M}$ 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.

• Countably generated space – topological space in which the topology is determined by its countable subsetsPages displaying wikidata descriptions as a fallback

• Bourbaki, Topologie générale, 2007.
• Bourbaki, Nicolas (1989) [1966]. 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