direct image
Let $f:A\u27f6B$ 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 $f({U}^{\mathrm{\complement}})\supset f(A)\setminus f(U)$.

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^{}.
Title  direct image 
Canonical name  DirectImage 
Date of creation  20130322 11:52:01 
Last modified on  20130322 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 8100 
Classification  msc 1800 
Classification  msc 17B37 
Classification  msc 18D10 
Classification  msc 18D35 
Classification  msc 16W30 
Synonym  image 
Related topic  InverseImage 
Related topic  Mapping 