The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of:


Non trivial vector bundles over a paracompact non-Hausdorff contractible space


Non trivial vector bundles over a Hausdorff non-paracompact contractible space


Non trivial vector bundles over a non-Hausdorff non-paracompact contractible space



Non trivial vector bundle over non-paracompact contractible space
Let $U$ and $V$ be two copies of the real line and make a space $X$ by gluing them by the identity along the strictly positive half-line: $x\in U$ equals $x\in V$ for $x>0$. Now make a rank one vector bundle over this space by taking a trivial bundle over each of the lines: glue $U\times\mathbb R$ to $V\times \mathbb R$ by identifying $(x,y)\in U\times\mathbb R$ with $(x,f(x)y)\in V\times\mathbb R$ for $x>0$. Let's choose the clutching function to be $f(x)=x$, or any other nowhere zero continuous function of positive real $x$ that cannot be extended to a nowhere zero function of all real $x$. Surely $X$ is contractible, but the bundle can't be trivial since the function $f:U\cap V\to GL\_1(\mathbb R)$ cannot be expressed as the quotient of a function extendible to $U$ and a function extendible to $V$, since that would make it itself extendible to all of $\mathbb R$.


