Research of applications of set theory to the,extension problem of continuous functions

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1. A topological space X is said to have the property (C^* =C) if every C^*-embedded closed set of X is C-embedded. In the extension theory of continuous functions, it is an interesting problem to construct a topological space, preferably a completely regular space satisfying the first axiom of countability, which fails to have the property (C^* = C). Concerning this problem, the following results are proved :(1) There exists a completely regular space X in which every point is a G-delta, and which does not have the property (C^*=C).(2) The Stone space of the Boolean algebra consisting of regular open sets of a completely regular space which is not weakly normal fails to have the property (C^* = C).(3) The Niemytzki plane has the property (C^* = C).2. Let C(Y) denote the Banach space of all continuous functions on a locally compact space Y which vanish at infinity. We define upper (lower) semi-continuity of maps to C(Y) and establish certain duality between upper (lower) semi-continuity of maps to C(Y) and upper (lower) semi-continuity of set-valued maps to the space of compact sets of C(Y). As an application, the following results are obtained :(1) A Hausdorff space X is paracompact if and only if for every locally compact space Y and every upper semi-continuous map f ; X → C(Y), there exists a continuous map g : X → C(Y) such that f(x) < g(x) for each x in X.(2) Every point-finite open cover of a topological space X is normal if and only if for every discrete space Y and every two maps g and h from X to C(Y) such that g is upper semi-continuous, h is lower semi-continuous and g 【less than or equal】 h, there exists a continuous map f such that g 【less than or equal】 f 【less than or equal】 h.