Katsuro Sakai, Shigenori Uehara
Topological structure of the space of lower semi-continuous functions

Comment.Math.Univ.Carolinae 47,1 (2006) 113-126.

Abstract:Let $L(X)$ be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space $X$, where, by identifying each $f$ with the epi-graph $epi(f)$, $L(X)$ is regarded the subspace of the space $Cld^*_F(X \times \Bbb R)$ of all closed sets in $X \times \Bbb R$ with the Fell topology. Let $$ \gather LSC(X) = \{f\in L(X) \mid f(X) \cap \Bbb R \not =\emptyset , f(X)\subset (-\infty ,\infty ]\} \text { and} LSC_{B}(X) = \{f \in L(X) \mid f(X) \text { is a bounded subset of $\Bbb R$}\}. \endgather $$ We show that $L(X)$ is homeomorphic to the Hilbert cube $Q = [-1,1]^\Bbb N$ if and only if $X$ is second countable, locally compact and infinite. In this case, it is proved that $(L(X), LSC(X), LSC_{B}(X))$ is homeomorphic to $(Cone Q, Q\times (0,1), \Sigma \times (0,1))$ (resp. $(Q,s,\Sigma )$) if $X$ is compact (resp. $X$ is non-compact), where $Cone Q = (Q \times \bold I)/(Q\times \{1\})$ is the cone over $Q$, $s = (-1,1)^\Bbb N$ is the pseudo-interior, $\Sigma = \{(x_i)_{i\in \Bbb N} \in Q \mid \sup _{i\in \Bbb N}|x_i| < 1\}$ is the radial-interior.

Keywords: space of lower semi-continuous functions, epi-graph, Fell topology, Hilbert cube, pseudo-interior, radial-interior
AMS Subject Classification: 57N20, 54C35

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