Let $V$ be a vector space.

Suppose we are given some upper semi-continuous map $\pi:V\rightarrow \bigcup_{k\le d} Gr(V,d)$, i.e, for any $x\in V$ we specify some subspace $\pi(x) \subseteq V$ of dimension no more than $d$, and these vary in a upper semi-continuous manner.

Must there exist a function $\eta$ from $V$ to the grassmannian $Gr(V,d)$ which is continuous/smooth, so that $\eta(x)$ always contains $\pi(x)$ as a subset?

these vary in a upper semi-continuous manner? As far I know semi-continuity refers only to real valued maps. en.wikipedia.org/wiki/Semi-continuity $\endgroup$ – Liviu Nicolaescu Sep 10 '16 at 13:15