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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$,
    $\displaystyle 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$,
    $\displaystyle 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$,
    $\displaystyle 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$,
    $\displaystyle f^{-1}(f(U)) \supset U$
    with equality if $ f$ is injective.
  6. Direct image of an inverse image: For any $ V \subset B$,
    $\displaystyle f(f^{-1}(V)) \subset V$
    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 157 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 12225 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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