Let X is a locally compact Hausdorff space, given an vector bundle p: E$\to$X, a subspace Y of X is called trivial (for this bundle), if we restrict this bundle over Y, it is a trivial bundle. In other word, p: p$^{-1}$(Y)$\to$Y is a trivial bundle. Y is called maximal trivial, if it is trivial and there is no trivial subspace of X strictly containing Y.
Given a point x in X, the maximal trivial subspace containing x maybe not unique, what can we say about them? Does any maximal trivial subspace must be open?
Let $X = S^1$. It admits a nontrivial real line bundle. On the other hand, since $S^1$ minus any point is contractible, every proper subspace of $S^1$ is trivial. Hence the maximal trivial subspaces are precisely the complements of points.