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

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

Let X be a hausdorff space such that every real vector bundle on X is summand of a trivial bundle. Does it implies 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

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