# direct image

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. 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. 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. 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. 4.

Subsets: If $U\subset V\subset A$, then $f(U)\subset f(V)\subset B$.

5. 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. 6.

Direct image of an inverse image: For any $V\subset B$,

 $f(f^{-1}(V))\subset V$

with equality if $f$ is surjective.

 Title direct image Canonical name DirectImage Date of creation 2013-03-22 11:52:01 Last modified on 2013-03-22 11:52:01 Owner djao (24) Last modified by djao (24) Numerical id 10 Author djao (24) Entry type Definition Classification msc 03E20 Classification msc 81-00 Classification msc 18-00 Classification msc 17B37 Classification msc 18D10 Classification msc 18D35 Classification msc 16W30 Synonym image Related topic InverseImage Related topic Mapping