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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 $USC(X,[0,1])$ ?
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  • $\begingroup$ topology of pointwise convergence $\endgroup$
    – Alexander Osipov
    Commented Mar 4, 2018 at 20:26
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    $\begingroup$ 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? $\endgroup$
    – Taras Banakh
    Commented Mar 4, 2018 at 23:39
  • $\begingroup$ The question can be narrowed down to perfectly normal spaces $X$ with a countable injective weight. $\endgroup$
    – Alexander Osipov
    Commented Mar 5, 2018 at 2:47
  • $\begingroup$ What about metrizable compact spaces $X$? $\endgroup$
    – Taras Banakh
    Commented Mar 5, 2018 at 5:59
  • $\begingroup$ 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. $\endgroup$
    – Alexander Osipov
    Commented Mar 5, 2018 at 7:49

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