Locally closed subset (original) (raw)

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In topology, a branch of mathematics, a subset E {\displaystyle E} {\displaystyle E} of a topological space X {\displaystyle X} {\displaystyle X} is said to be locally closed if any of the following equivalent conditions are satisfied:[1][2][3][4]

The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed.[1] To see the second condition implies the third, use the facts that for subsets A ⊆ B , {\displaystyle A\subseteq B,} {\displaystyle A\subseteq B,} A {\displaystyle A} {\displaystyle A} is closed in B {\displaystyle B} {\displaystyle B} if and only if A = A ¯ ∩ B {\displaystyle A={\overline {A}}\cap B} {\displaystyle A={\overline {A}}\cap B} and that for a subset E {\displaystyle E} {\displaystyle E} and an open subset U , {\displaystyle U,} {\displaystyle U,} E ¯ ∩ U = E ∩ U ¯ ∩ U . {\displaystyle {\overline {E}}\cap U={\overline {E\cap U}}\cap U.} {\displaystyle {\overline {E}}\cap U={\overline {E\cap U}}\cap U.}

The interval ( 0 , 1 ] = ( 0 , 2 ) ∩ [ 0 , 1 ] {\displaystyle (0,1]=(0,2)\cap [0,1]} ![{\displaystyle (0,1]=(0,2)\cap [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/157cfdd164e17933e78d6db1f65b1a67fa577766) is a locally closed subset of R . {\displaystyle \mathbb {R} .} {\displaystyle \mathbb {R} .} For another example, consider the relative interior D {\displaystyle D} {\displaystyle D} of a closed disk in R 3 . {\displaystyle \mathbb {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)\}} {\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}} {\displaystyle \mathbb {R} ^{2}}.

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

  1. ^ a b c Bourbaki 2007, Ch. 1, § 3, no. 3.
  2. ^ a b c Pflaum 2001, Explanation 1.1.2.
  3. ^ Ganster, M.; Reilly, I. L. (1989). "Locally closed sets and LC -continuous functions". International Journal of Mathematics and Mathematical Sciences. 12 (3): 417–424. doi:10.1155/S0161171289000505. ISSN 0161-1712.
  4. ^ Engelking 1989, Exercise 2.7.1.
  5. ^ 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.section 1, p. 476
  6. ^ Bourbaki 2007, Ch. 1, § 3, Exercise 7.