I am interested in the topological homogeneity of function spaces.

Question. Let $X$ be a Tychonoff space, let $USC(X)$ be a space of upper semicontinuous functions on $X$ and let $USC(X)^+$ be a space of non-negative upper semicontinuous functions on $X$.

- Is the space $USC(X)$ topologically homogeneous ?
- Is the space $USC(X)^+$ topologically homogeneous ?
- What about $USC(X,[0,1])$ ?