Spaces over which every vector bundle is a summand of the trivial bundle Let X be  a  Hausdorff space such that every real  vector  bundle on X is  summand of  a trivial bundle. Does  this imply that X is  homotopy  equivalent to a  compact  Hausdorf space? This  question is  a "compact  version" of the  following question;
Paracompactness and inner product on vector bundles
 A: Let $X$ be the wedge of infinitely many circles (equipped with the CW topology). Every vector bundle $\xi$ over $X$ is a summand of a trivial bundle, namely it is $\xi\oplus\xi$ is trivial because any vector bundle over a circle has this property (alternatively, one could appeal to the fact that $X$ is homotopy equivalent to a smooth manifold, if the number of circles is countable, as explained in my comment above). 
Suppose there is a compact space  $K$ and a homotopy equivalence $f: K\to X$. The image $f(K)$ is compact, so $f(K)$ lies in a finite subcomplex $X_0$ of $X$, i.e. $X_0$ is a wedge of finitely many circles. Thus any loop on $X$ is freely homotopic to a loop in $X_0$, which is false (because a circle that forms the wedge
can be homotoped into $X_0$ only if it lies in $X_0$.
EDIT: A key feature of the above example is that there is a homotopic to the identity $X\to X$ whose image lies in a compact subset, and it allows for the following optimal generalization. 
Let $X$ be a (say path-connected) space homotopy equivalent to a locally compact ANR (such as a manifold, or a locally finite CW complex) which we denote $\bar X$. Suppose $X$ is homotopy equivalent to a compact space. Then there is a homotopic to the identity map $\bar X\to \bar X$ whose image lies in a compact set. By a standard argument (see e.g. proposition 3.18 of http://arxiv.org/abs/1210.6741
of Guilbault's survey), this is equivalent to assuming that $\bar X$ is finitely dominated (by a finite CW complex). Note that finitely dominated spaces have finitely generated homology groups, and finitely presented fundamental groups (see e.g proposition 3.16 in the above paper). Of course, the fundamental group of a wedge of infinitely many circles is not finitely presented. 
Conversely, Ferry proved in "Homotopy, simple homotopy, and compacta" (Topology 9 (1980) pp 101-110) that any space that is dominated by a compact Hausdorff space is homotopy equivalent to a compact Hausdorff space. Thus we conclude that a locally compact ANR is finitely dominated if and only if it is homotopy equivalent to a compact Hausdorff space.
In particular,  if $X$ is homotopy equivalent to a smooth manifold that is not finitely dominated, then $X$ is not homotopy equivalent to a compact Hausdorff space, even though any vector bundle over $X$ is a summand of the trivial bundle.
