| Title:
|
Topological structure of the space of lower semi-continuous functions (English) |
| Author:
|
Sakai, Katsuro |
| Author:
|
Uehara, Shigenori |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
113-126 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
space of lower semi-continuous functions |
| Keyword:
|
epi-graph |
| Keyword:
|
Fell topology |
| Keyword:
|
Hilbert cube |
| Keyword:
|
pseudo-interior |
| Keyword:
|
radial-interior |
| MSC:
|
54C35 |
| MSC:
|
57N20 |
| idZBL:
|
Zbl 1150.57006 |
| idMR:
|
MR2223971 |
| . |
| Date available:
|
2009-05-05T16:56:00Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119578 |
| . |
| Reference:
|
[1] Anderson R.D.: On sigma-compact subsets of infinite-dimensional spaces.unpublished. |
| Reference:
|
[2] Beer G.: Topologies on Closed and Closed Convex Sets.Math. and its Appl. 268, Kluwer Acad. Publ., Dordrecht, 1993. Zbl 0792.54008, MR 1269778 |
| Reference:
|
[3] Chapman T.A.: Dense sigma-compact subsets in infinite-dimensional manifolds.Trans. Amer. Math. Soc. 154 (1971), 399-426. MR 0283828 |
| Reference:
|
[4] Curtis D.W.: Boundary sets in the Hilbert cube.Topology Appl. 20 (1985), 201-221. Zbl 0575.57008, MR 0804034 |
| Reference:
|
[5] Fell J.M.G.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space.Proc. Amer. Math. Soc. 13 (1962), 472-476. Zbl 0106.15801, MR 0139135 |
| Reference:
|
[6] Kubiś W., Sakai K., Yaguchi M.: Hyperspaces of separable Banach spaces with the Wijsman topology.Topology Appl. 148 (2005), 7-32. Zbl 1068.54011, MR 2118072 |
| Reference:
|
[7] Lawson J.D.: Topological semilattices with small subsemilattices.J. London Math. Soc. (2) 1 (1969), 719-724. MR 0253301 |
| Reference:
|
[8] van Mill J.: Infinite-Dimensional Topology, Prerequisites and Introduction.North-Holland Math. Library 43, Elsevier Sci. Publ. B.V., Amsterdam, 1989. Zbl 0663.57001, MR 0977744 |
| Reference:
|
[9] Sakai K., Yang Z.: Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube.Topology Appl. 127 (2002), 331-342. MR 1941172 |
| Reference:
|
[10] Toruńczyk H.: On CE-images of the Hilbert cube and characterization of $Q$-manifolds.Fund. Math. 106 (1980), 31-40. MR 0585543 |
| . |
