On 𝑆-closed spaces

Abstract

In this paper, we initially give several new characterizations of the class of <italic>S</italic> -closed spaces, which was introduced by T. Thompson [Proc. Amer. Math. Soc. <bold>60</bold> (1976), 335-338]. We then employ these characterizations to produce analogues for <italic>S</italic> -closed spaces of the well-known theorem from real analysis that an upper-semicontinuous real-valued function on a closed interval assumes a maximum, 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 <italic>S</italic> -closed spaces. A “fixed set theorem” for multifunctions on <italic>S</italic> -closed spaces is also established.