In topology, a branch of mathematics, a subset E {\displaystyle E} of a topological space X {\displaystyle X} is said to be locally closed if any of the following equivalent conditions are satisfied:
E {\displaystyle E} is the intersection of an open set and a closed set in X . {\displaystyle X.}
For each point x ∈ E , {\displaystyle x\in E,} there is a neighborhood U {\displaystyle U} of x {\displaystyle x} such that E ∩ U {\displaystyle E\cap U} is closed in U . {\displaystyle U.}
E {\displaystyle E} is open in its closure E ¯ . {\displaystyle {\overline {E}}.}
The set E ¯ ∖ E {\displaystyle {\overline {E}}\setminus E} is closed in X . {\displaystyle X.}
E {\displaystyle E} is the difference of two closed sets in X . {\displaystyle X.}
E {\displaystyle E} is the difference of two open sets in X . {\displaystyle 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 ⊆ B , {\displaystyle A\subseteq B,} A {\displaystyle A} is closed in B {\displaystyle B} if and only if A = A ¯ ∩ B {\displaystyle A={\overline {A}}\cap B} and that for a subset E {\displaystyle E} and an open subset U , {\displaystyle U,} E ¯ ∩ U = E ∩ U ¯ ∩ U . {\displaystyle {\overline {E}}\cap U={\overline {E\cap U}}\cap U.}
Examples The interval ( 0 , 1 ] = ( 0 , 2 ) ∩ [ 0 , 1 ] {\displaystyle (0,1]=(0,2)\cap [0,1]} is a locally closed subset of R . {\displaystyle \mathbb {R} .} For another example, consider the relative interior D {\displaystyle D} of a closed disk in R 3 . {\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, { ( x , y ) ∈ R 2 ∣ x ≠ 0 } ∪ { ( 0 , 0 ) } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\neq 0\}\cup \{(0,0)\}} is not a locally closed subset of R 2 {\displaystyle \mathbb {R} ^{2}} . Recall that, by definition, a submanifold E {\displaystyle E} of an n {\displaystyle n} -manifold M {\displaystyle M} is a subset such that for each point x {\displaystyle x} in E , {\displaystyle E,} there is a chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} around it such that φ ( E ∩ U ) = R k ∩ φ ( U ) . {\displaystyle \varphi (E\cap U)=\mathbb {R} ^{k}\cap \varphi (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 ∩ Y ¯ {\displaystyle Y=U\cap {\overline {Y}}} where Y ¯ {\displaystyle {\overline {Y}}} denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)
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