Epi-拓扑的某些性质

Throughout this paper R denotes the set of all real numbers.An extended real valued function j f X-[--oo, +oc] is called lower semicontinuouson X if fOr each real Q, {x' f(x) > o} is open in X. Clearly, f is lower semicontinuous ifand only if, its epigraph eghf = {(x, a) f f(x) 5 cr} is closed in X x R. f is caJ1ed proper ifit is somewhere finite, and its values lie in (--co, +oc]. We denote by LSC(X) the set of ajlproper lower semicontinuous functions.In the sequel, we identify lower se…

Some Properties of the Epi-Topology

Let LSC(X) denote the set of all proper lower semicontinuous functions on X with the epi-topology. In this paper we give characterizations of the separation axioms. Baire properties and metrizability of LSC(X). We show also that the continuous function space C(X) with the epi-topology is of first category when N is first countable.