Editing Sheaf (mathematics) (section)

==== Sheaf of sections of a continuous map ====
Any continuous map <math>f:Y\to X</math> of topological spaces determines a sheaf <math>\Gamma(Y/X)</math> on <math>X</math> by setting
:<math>\Gamma(Y/X)(U) = \{s: U \to Y, f \circ s = \operatorname{id}_U\}.</math>
Any such <math>s</math> is commonly called a [[section (category theory)|section]] of ''<math>f</math>'', and this example is the reason why the elements in <math>\mathcal{F}(U)</math> are generally called sections. This construction is especially important when <math>f</math> is the projection of a [[fiber bundle]] onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the [[trivial bundle]]. 

Another example: the sheaf of sections of
:<math>\C \stackrel{\exp}{\longrightarrow} \C\setminus \{0\}</math>
is the sheaf which assigns to any ''<math>U\subseteq \mathbb{C}\setminus\{0\}</math>'' the set of branches of the [[complex logarithm]] on ''<math>U</math>''.

Given a point <math>x</math> and an abelian group <math>S</math>, the skyscraper sheaf <math>S_x</math> is defined as follows: if <math>U</math> is an open set containing <math>x</math>, then <math>S_x(U)=S</math>. If <math>U</math> does not contain <math>x</math>, then <math>S_x(U)=0</math>, the [[trivial group]]. The restriction maps are either the identity on <math>S</math>, if both open sets contain <math>x</math>, or the zero map otherwise.