Topological problems associated with \(\Gamma\)-convergence (Q793972)

If X is a Hausdorff topological space and S(X) denotes the class of all lower semicontinuous functions \(f:X\to {\bar {\mathbb{R}}},\) then the \(\Gamma\)-convergence (for the definitions see \textit{E. De Giorgi, T. Franzoni} [Atti Accad. Naz. Lincei, VIII Ser., Rend., Cl. Sci. Fis. Mat. Natur, 58, 842-850 (1975; Zbl 0339.49005)] and \textit{G. Buttazzo} [Boll. Unione Mat. Ital., V. Ser., B 14, 722-744 (1977; Zbl 0445.49016)] is a convergence for sequences, and more generally for nets, on S(X). In this paper, some topologies related to the definition of the \(\Gamma\)- convergence are introduced. More precisely, the topologies \(\tau_{seq}\) and \(\tau_{net}\) are defined as the strongest topologies on S(X) for which all \(\Gamma\)-convergent sequences (respectively nets) are convergent. A third topology \(\tau_{\min}\) is defined as the topology generated by the sets \(\{f\in S(X):\inf_{A} f<t\}\) and \(\{f\in S(X):\inf_{K} f>s\}\) for s,\(t\in {\mathbb{R}}\), A open subset of X, K compact subset of X. The properties of these topologies and their relations with the \(\Gamma\)-convergence are examined: in particular, if X is an infinite dimensional separable Hilbert space, then for every topology \(\tau_{seq}\), \(\tau_{net}\), \(\tau_{\min}\) we have that all non empty open subsets of S(X) are dense in S(X). Some of the properties studied here, have been useful in problems like stochastic homogenization [see \textit{G. Facchinetti} and \textit{L. Russo}, ibid. VI. Ser., C 2, 159- 170 (1983) and the author and L. Modica (preprint)] in which a sequence of probability measures on S(X) is considered.