A section of a locally trivial algebra bundle (E,p,B) is a continuous map $S$ from $p:B\to E$ such $p\circ S= ID$. For an ideal bundle $H$ of $E$, I am trying to construct sections of $E$ such that every element of $H_x$ is in some section of $E$.
I have listed my argument here.
Suppose my base space is compact Hausdorff. Then $B$ is a union of connected components and fibres are isomorphic on each component.Now for a particular $x\in B$, and $y \in H_x$ I set, $S(x)=y$ throughout the component containing $x$. On all other components I fix a particular element of the fibre. Then on each component S is continuous and by pasting lemma, S is a section of $E$. This argument looks fine but still is not convincing to me. Can anyone help me to refine it?