direct imageDirectImage2001-10-21 01:46:422005-07-24 03:33:50Definition\usepackage{amssymb}
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\usepackage{xypic}Let $f\colon A \longrightarrow B$ be a function, and let $U \subset A$ be a subset. The {\em 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:
\begin{enumerate}
\item Unions: For any collection $\{U_i\}_{i \in I}$ of subsets of $A$,
$$
f\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i\in I} f(U_i).
$$
\item Intersections: For any collection $\{U_i\}_{i \in I}$ of subsets of $A$,
$$
f\left(\bigcap_{i \in I} U_i\right) \subset \bigcap_{i\in I} f(U_i).
$$
\item 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 $f(U^\complement) \supset f(A) \setminus f(U)$.
\item Subsets: If $U \subset V \subset A$, then $f(U) \subset f(V) \subset B$.
\item Inverse image of a direct image: For any $U \subset A$,
$$f^{-1}(f(U)) \supset U$$
with equality if $f$ is injective.
\item Direct image of an inverse image: For any $V \subset B$,
$$f(f^{-1}(V)) \subset V$$
with equality if $f$ is surjective.
\end{enumerate}