# Homogeneity of the space of semicontinuous functions

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 $X$.

1. Is the space $USC(X)$ topologically homogeneous ?
2. Is the space $USC(X)^+$ topologically homogeneous ?
3. What about $USC(X,[0,1])$ ?
• topology of pointwise convergence – Alexander Osipov Mar 4 '18 at 20:26
• I think that for a countable Tychonoff space containing a non-trivial convergent sequence it is possible to prove that all three spaces are absorbing and hence topologically homogeneous. Is such answer satisfactory for your purposes? – Taras Banakh Mar 4 '18 at 23:39
• The question can be narrowed down to perfectly normal spaces $X$ with a countable injective weight. – Alexander Osipov Mar 5 '18 at 2:47
• What about metrizable compact spaces $X$? – Taras Banakh Mar 5 '18 at 5:59
• Taras, even for a metrizable compact, I do not know the answer. But great interest for perfectly normal spaces with a countable i-weight of power is no more than a continuum. – Alexander Osipov Mar 5 '18 at 7:49