Topological structure of the space of lower semi-continuous functions

Sakai, Katsuro ; Uehara, Shigenori

Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 1, pp. 113-126.

Résumé

Let $\operatorname{L}(X)$ 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 $\operatorname{epi}(f)$, $\operatorname{L}(X)$ is regarded the subspace of the space $\operatorname{Cld}^*_F(X \times \Bbb R)$ of all closed sets in $X \times \Bbb R$ with the Fell topology. Let $$ \operatorname{LSC}(X) = \{f\in \operatorname{L}(X) \mid f(X) \cap \Bbb R \neq \emptyset, f(X)\subset (-\infty,\infty]\} \text{ and} \ \operatorname{LSC}_{\operatorname{B}}(X) = \{f \in \operatorname{L}(X) \mid f(X) \text{ is a bounded subset of $\Bbb R$}\}. $$ We show that $\operatorname{L}(X)$ 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 $(\operatorname{L}(X), \operatorname{LSC}(X), \operatorname{LSC}_{\operatorname{B}}(X))$ is homeomorphic to $(\operatorname{Cone} Q, Q\times (0,1), \Sigma \times (0,1))$ (resp. $(Q,s,\Sigma)$) if $X$ is compact (resp. $X$ is non-compact), where $\operatorname{Cone} Q = (Q \times \bold I)/(Q\times \{1\})$ 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.

Détail

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