Closed filters and graph-closed multifunctions in convergence spaces (Q1101988)

Every graph-closed multifunction \(\Gamma\) : \(Y\to X\) is closed-valued, has a closed-valued inverse \(\Gamma^{-1}\) and \(\Gamma\) (k) is closed whenever k is a compact subset of Y. However, in general compact-to- closed multifunctions need not be graph-closed. The equivalence may be obtained under the assumption that Y is a locally compact Hausdorff space [see \textit{R. E. Smithson}, Pac. J. Math. 61, 283-288 (1975; Zbl 0317.54022)]. It follows from \textit{S. Mrowka} [Lect. Notes Math. 171, 59- 63 (1970; Zbl 0215.516)] that if Y is Hausdorff and every multifunction \(\Gamma\) : \(Y\to X\) with a closed-valued inverse \(\Gamma^{-1}\), the equivalence holds, then Y is locally compact. The author shows that graph-closedness can be expressed in terms of some corresponding properties of images of filters. Namely, graph-closed multifunctions are exactly those multifunctions with closed-valued inverses that map compact filters into closed filters.