Equilibria of nonparacompact generalized games with \(\mathcal L_c\)-majorized correspondences in \(FC\)-spaces (Q2518128)

The referred paper deals with a topological space \(Y\) which is equipped with an indexed family of continuous functions from standard simplexes into \(Y\) as follows. The family of indexes consists of all non-empty and finite subsets \(Y\). A continuous map \(\phi_N: \Delta_n \to Y\) is associated with every index \(N=\{y_0, y_1, y_2, \dots , y_n\}\). The pair \((Y, \{\phi_N\})\) is -- inconsistently with usual set theoretical or topological conventions -- called a finitely continuous topological space or \(FC\)-space. The assumption that elements of every index are numbered is of vital importance for the next definition. A subset \(D\subseteq Y\) is called an \(FC\)-subspace whenever for each index \(N\) and for any \(K\subseteq N\cup D\) the image \(\phi_N [\Delta_K]\) is contained in \(D\). Here \(\Delta_K \) is the \(|K|\)-dimensional face with vertices \(\{e_i: y_i \in K\}\) of the the \(n\)-dimensional simplex \(\Delta_n\) with vertices \(e_0, e_1, \dots, e_n\). If \(X\subseteq Y\), then the intersection of all \(FC\)-subspaces \(B \supseteq X\) is called the \(FC\)-hull of \(X\). The authors declare that using the ``new'' concept of \(FC\)-hull they unify, improve and generalize many results about topological vector spaces, \(H\)-spaces or \(G\)-convex spaces. At last, one can say: Whenever \( \phi_N [\Delta_K]\subseteq \phi_K [\Delta_k]\) is assumed for \(K \subseteq N\) in the definition of \(FC\)-spaces, then we have the definition of a \(L\)-structure on \(Y\) (\(L\)-space) which was introduced in the paper [\textit{H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano} and \textit{J.-V. Llinares}, J. Math. Anal. Appl. 222, No.~1, 138--150 (1998; Zbl 0986.54054)]. Alternatively, \( \phi_N [\Delta_K] = \phi_K [\Delta_k]\) gives the definition of a simplicial structure in \(Y\) [see \textit{W. Kulpa}, Topol. Proc. 22, 211--235, electronic only (1997; Zbl 0948.47057)]. \textit{S. Park} says that the notion of \(G\)-convex space (but with some additional isotonicity condition which was rejected in 1988) was introduced in his paper (joint with \textit{H. Kim}) ``Admissible classes of multifunctions on generalized convex spaces''. The definition of \(G\)-convex space is as the definition of \(L\)-space, but a family of indexes consists of all non-empty and finite subsets of a fixed set, not necessarily \(Y\). Notions of simplicial structure, \(L\)-space, \(G\)-convex space or \(FC\)-space are frameworks which are based on combinatorial methods introduced by \textit{E. Sperner} in the paper [Abhandlungen Hamburg 6, 265--272 (1928; JFM 54.0614.01)]. Up to date, there are no counterexamples which clearly show differences between results in accordance with frameworks. Albeit, frameworks of simplicial structure or \(L\)-space simplify the \(G\)-convex framework. The \(FC\)-spaces framework is oversimplification and not convincing. It was introduced a few years later and it shifts some assumptions, only.