Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called *trivialising* (for this bundle), if after restricting this bundle to $Y$, it is a trivial bundle. In other words, $p: p^{-1}(Y)\to Y$ is trivial. $Y$ is called *maximally trivial*, if it is trivial and there is no trivial subspace of $X$ which contains $Y$ as a proper subset.

Given a point $x$ in $X$, a maximal trivialising subspace containing $x$ may not be unique. Does any maximal trivialising subspace must be open? Can we say something others about maximal trivialising subspaces?

There is no reference about this topic in the existing literatures.

A similar question has been posted on stackexchange: https://math.stackexchange.com/questions/1581861/maximal-trivial-subspace-in-vector-bundles