topos theory

# Contents

## Definition

Given a sheaf $F$ of sets over a topological space $X$, the section over an arbitrary (= not necessarily open) subset $V\subset X$ is a continuous section of the corresponding etale space restricted to $V$.

A sheaf $F$ of sets (or of abelian groups) over a paracompact Hausdorff space $X$ is soft if for any closed subset $V\subset X$, every section of $F$ over $V$ can be extended to the whole $X$.

## Properties

The local extension of germs to the open neighborhoods of points by paracompactness gives rise to an extension of the section to an open neighborhood of the whole set $V$. Therefore every flabby sheaf is soft, because flabbiness gives the extension from open subsets. Fine sheaves are always soft.

## References

Standard references are Tohoku and Godement’s book.

category: sheaf theory

Revised on March 6, 2013 19:50:09 by Zoran Škoda (161.53.130.104)