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Abstract. In this paper, we initially give several new characterizations of the class of 5-closed spaces, which was introduced by T. Thompson [Proc. Amer. Math. Soc. 60 (1976), 335-338]. We then employ these characterizations to produce analogues for 5-closed spaces of the well-known theorem from real analysis that an uppersemicontinuous real-valued function on a closed interval assumes a maTrimnm, and of two well-known theorems of G. Birkhoff and A. D. Wallace, which established that each upper-semicontinuous function from a compact space into a partially ordered set assumes a maximal value and that each compact space has a maximal element with respect to each upper-semicontinuous quasi order on the set. The statements in these latter analogues are then shown to characterize 5-closed spaces. A "fixed set theorem" for multifunctions on 5-closed spaces is also established.

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