Locally closed subset

 ( General Topology ) 

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 $x\backslash in\; E,$ there is a neighborhood $U$ of $x$ such that $E\; \backslash cap\; U$ is closed in $U.$
• $E$ is open in its closure $\backslash overline\{E\}.$
• The set $\backslash overline\{E\}\backslash setminus\; E$ 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 $A\; \backslash subseteq\; B,$ $A$ is closed in $B$ if and only if $A\; =\; \backslash overline\{A\}\; \backslash cap\; B$ and that for a subset $E$ and an open subset $U,$ $\backslash overline\{E\}\; \backslash cap\; U\; =\; \backslash overline\{E\; \backslash cap\; U\}\; \backslash cap\; U.$

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

On the other hand, $\backslash \{\; (x,y)\backslash in\backslash Reals^2\; \backslash mid\; x\backslash ne0\; \backslash \}\; \backslash cup\; \backslash \{(0,0)\backslash \}$ is not a locally closed subset of $\backslash Reals^2$.

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 $\backslash varphi\; :\; U\; \backslash to\; \backslash Reals^n$ around it such that $\backslash varphi(E\; \backslash cap\; U)\; =\; \backslash Reals^k\; \backslash cap\; \backslash varphi(U).$ Hence, a submanifold is locally closed.section 1, p. 476

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, $Y\; =\; U\; \backslash cap\; \backslash overline\{Y\}$ where $\backslash overline\{Y\}$ denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)

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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 $\backslash overline\{E\}\; \backslash setminus\; E$ 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.

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See also

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Notes

• (2026). 9783540339823, Springer. . ISBN 9783540339823
• Ryszard Engelking (1989). 9783885380061, Heldermann Verlag, Berlin. ISBN 9783885380061
• Markus J. Pflaum (2026). 9783540426264, Springer. ISBN 9783540426264

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External links

Categories: General Topology

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