Posted: Tue Oct 26, 2004 3:32 pm    Post subject: Upper-hemi-continuous Correspondence

A correspondence is a “rule” which associates to every element a non-empty subset .

A correspondence is said to be upper-hemi- continuous(u.h.c.) at if for every open set containing there exists a neighbourhood of such that for every

Prove the following assertion:

If the correspondence is u.h.c. at x, then the closure of , i.e. the correspondence is u.h.c.

Posted: Tue Oct 26, 2004 8:11 pm    Post subject:

One more question: How to prove the following proposition?? (Carathedory):

Let be a subset of . Then every point is the convex combination of some points in , where is the convex hull of (i.e. the smallest convex set containing )

Posted: Tue Nov 16, 2004 9:52 pm    Post subject:

For the first question, if there any extra information about the topological spcaces S and T ?

I can prove under an extra assumption that T is compact Hausdorff.