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direct image (Definition)

Let $f\colon A \longrightarrow B$ be a function, and let $U \subset A$ be a subset. The direct image of $U$ is the set $f(U) \subset B$ consisting of all elements of $B$ which equal $f(u)$ for some $u \in U$

Direct images satisfy the following properties:

  1. Unions: For any collection $\{U_i\}_{i \in I}$ of subsets of $A$ $$ f\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i\in I} f(U_i). $$
  2. Intersections: For any collection $\{U_i\}_{i \in I}$ of subsets of $A$ $$ f\left(\bigcap_{i \in I} U_i\right) \subset \bigcap_{i\in I} f(U_i). $$
  3. Set difference: For any $U,V \subset A$ $$ f(V \setminus U) \supset f(V) \setminus f(U). $$ In particular, the complement of $U$ satisfies $f(U^\complement) \supset f(A) \setminus f(U)$
  4. Subsets: If $U \subset V \subset A$ then $f(U) \subset f(V) \subset B$
  5. Inverse image of a direct image: For any $U \subset A$ $$f^{-1}(f(U)) \supset U$$ with equality if $f$ is injective.
  6. Direct image of an inverse image: For any $V \subset B$ $$f(f^{-1}(V)) \subset V$$ with equality if $f$ is surjective.




"direct image" is owned by djao. [ full author list (2) ]
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See Also: inverse image, mapping

Other names:  image
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Cross-references: surjective, injective, equality, inverse image, complement, set difference, intersections, collection, unions, properties, satisfy, subset, function
There are 192 references to this entry.

This is version 5 of direct image, born on 2001-10-21, modified 2005-07-24.
Object id is 443, canonical name is DirectImage.
Accessed 14919 times total.

Classification:
AMS MSC03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

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