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Topological structure of the space of lower semi-continuous functions

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Katsuro Sakai, Shigenori Uehara

Topological structure of the space of lower semi-continuous functions

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

Abstract: LetL(X) be the space of all lower semi-continuous extended real-valued functions on a Hausdorff spaceX, where, by identifying eachf with the epi-graph epi(f),L(X) is regarded the subspace of the spaceCld∗F(X×R) of all closed sets inX×Rwith the Fell topology. Let

LSC(X) ={f ∈L(X)|f(X)∩R6=∅, f(X)⊂(−∞,∞]} andLSCB(X) ={f ∈L(X)|f(X) is a bounded subset ofR}.

We show thatL(X) is homeomorphic to the Hilbert cubeQ= [−1,1]N if and only if X is second countable, locally compact and infinite. In this case, it is proved that (L(X), LSC(X), LSCB(X)) is homeomorphic to (ConeQ, Q×(0,1),Σ×(0,1)) (resp. (Q, s,Σ)) if X is compact (resp. X is non-compact), where ConeQ = (Q×I)/(Q× {1}) is the cone over Q, s = (−1,1)N is the pseudo-interior, Σ = {(xi)i∈N∈Q|supi∈N|xi|<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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