Topological structure of the space of lower semi-continuous functions

Sakai, Katsuro, and Uehara, Shigenori. "Topological structure of the space of lower semi-continuous functions." Commentationes Mathematicae Universitatis Carolinae 47.1 (2006): 113-126. <http://eudml.org/doc/249835>.

@article{Sakai2006,
abstract = {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 $\operatorname\{Cld\}^*_F(X \times \mathbb \{R\})$ of all closed sets in $X \times \mathbb \{R\}$ with the Fell topology. Let \[ \operatorname\{LSC\}(X) = \lbrace f\in \operatorname\{L\}(X) \mid f(X) \cap \mathbb \{R\} \ne \emptyset , f(X)\subset (-\infty ,\infty ]\rbrace \text\{ and\} \ \operatorname\{LSC\}\_\{\operatorname\{B\}\}(X) = \lbrace f \in \operatorname\{L\}(X) \mid f(X) \text\{ is a bounded subset of $\mathbb \{R\}$\}\rbrace . \] We show that $\operatorname\{L\}(X)$ is homeomorphic to the Hilbert cube $Q = [-1,1]^\mathbb \{N\}$ 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 $\operatorname\{Cone\} Q = (Q \times \mathbf \{I\})/(Q\times \lbrace 1\rbrace )$ is the cone over $Q$, $s = (-1,1)^\mathbb \{N\}$ is the pseudo-interior, $\Sigma = \lbrace (x_i)_\{i\in \mathbb \{N\}\} \in Q \mid \sup _\{i\in \mathbb \{N\}\}|x_i| < 1\rbrace $ is the radial-interior.},
author = {Sakai, Katsuro, Uehara, Shigenori},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {space of lower semi-continuous functions; epi-graph; Fell topology; Hilbert cube; pseudo-interior; radial-interior; epi-graph; Fell topology; Hilbert cube},
language = {eng},
number = {1},
pages = {113-126},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Topological structure of the space of lower semi-continuous functions},
url = {http://eudml.org/doc/249835},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Sakai, Katsuro
AU - Uehara, Shigenori
TI - Topological structure of the space of lower semi-continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 1
SP - 113
EP - 126
AB - 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 $\operatorname{Cld}^*_F(X \times \mathbb {R})$ of all closed sets in $X \times \mathbb {R}$ with the Fell topology. Let \[ \operatorname{LSC}(X) = \lbrace f\in \operatorname{L}(X) \mid f(X) \cap \mathbb {R} \ne \emptyset , f(X)\subset (-\infty ,\infty ]\rbrace \text{ and} \ \operatorname{LSC}_{\operatorname{B}}(X) = \lbrace f \in \operatorname{L}(X) \mid f(X) \text{ is a bounded subset of $\mathbb {R}$}\rbrace . \] We show that $\operatorname{L}(X)$ is homeomorphic to the Hilbert cube $Q = [-1,1]^\mathbb {N}$ 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 $\operatorname{Cone} Q = (Q \times \mathbf {I})/(Q\times \lbrace 1\rbrace )$ is the cone over $Q$, $s = (-1,1)^\mathbb {N}$ is the pseudo-interior, $\Sigma = \lbrace (x_i)_{i\in \mathbb {N}} \in Q \mid \sup _{i\in \mathbb {N}}|x_i| < 1\rbrace $ is the radial-interior.
LA - eng
KW - space of lower semi-continuous functions; epi-graph; Fell topology; Hilbert cube; pseudo-interior; radial-interior; epi-graph; Fell topology; Hilbert cube
UR - http://eudml.org/doc/249835
ER -