direct image

Let $f:A⟶B$ 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 $f(U^{\mathrm{\complement }})\supset f(A)\setminus f(U)$.

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	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