Is the presheaf of continuous functions on a topological space a "complete presheaf"?

Question

Is the presheaf of continuous functions $f:A\rightarrow B$ from a topological space $A$ to another topological space $B$ a "complete presheaf"? Can't find this, anyone have a reference?

Answer 1

To check that the presheaf of continuous functions from $A$ to $B$ is a complete one it suffices to show that given an open cover $U = \bigcup_\alpha U_\alpha \subseteq A $ and continuous maps $f_\alpha : U_\alpha \rightarrow B$ which coincide on the intersections $U_\alpha \cap U_\beta$ there exists a unique map $f: U \rightarrow B$ which restricts to $f_\alpha$ on $U_\alpha$ (In the definition you linked, uniqueness and existence are the first and second property, respectively).

This can be done be defining $f(x) := f_\alpha(x)$ for $x \in U_\alpha$. First of all this is well defined because of the assumed compatibility of the $f_\alpha$ on the intersections and this is the only way we can hope to glue the $f_\alpha$ together. It remains to show continuity of $f$. Take $x \in U$ and an open neighborhood $V \subseteq B$ of $f(x)$. Then $x \in U_\alpha$ for some $\alpha$ and by continuity of of $f_\alpha$ there is an open $W \subset U_\alpha$ with $f_\alpha(W) \subseteq V$. It follows that $f(W) = f_\alpha(W) \subseteq V$ and hence $f$ is continuous.