Locally closed subset

In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if the following equivalent conditions are met:

• E is an intersection of an open subset and a closed subset of X.
• For each point x in E, there is a neighborhood U of x in X such that $E\cap U$ is closed in U.
• E is an open subset of the closure ${\overline {E}}$.

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 $A\subset B$, 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, ${\overline {E}}\cap U={\overline {E\cap U}}\cap U$.

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.)

For example, $(0,1]=(0,2)\cap [0,1]$ 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. 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 $\varphi :U\to \mathbb {R} ^{n}$ around it such that $\varphi (E\cap U)=\mathbb {R} ^{k}\cap \varphi (U)$. Hence, a submanifold is locally closed.

For a locally closed subset E, the complement ${\overline {E}}-E$ 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 (i.e., 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#S for more of this notion.

References

• Bourbaki, Topologie générale, 2007.
• 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.
• Markus J. Pflaum, Analytic and Geometric Study of Stratified Spaces: Contributions to Analytic and Geometric Aspects (Lecture Notes in Mathematics, 1768) ; Publisher, Springer;

Further reading

• https://ncatlab.org/nlab/show/locally+closed+set