(Quasi)-uniformities on the set of bounded maps (Q1338422)

If \(X\) is a set, and if \((Y,{\mathcal U})\), let \({\mathcal F} (X,Y)\) denote the set of all functions from \(X\) to \(Y\). The topology on \(Y\) generated by \({\mathcal U}\) is denoted by \({\mathcal T}_{\mathcal U}\). A subset \(A\) of \(Y\) is said to be \({\mathcal U}\)-bounded, if, given an entourage \(V \in {\mathcal U}\), there exists a positive integer \(n\) and a finite subset \(F\) of \(Y\) such that \(A \subset V^ n(F)\). Let \(\alpha\) be a collection of subsets covering \(X\), and let \[ \begin{aligned} B(X,Y) & = \{f \in {\mathcal F} (X,Y) : f(X) \quad \text{is } {\mathcal U} \text{-bounded}\} \\ B^*(X,Y) & = \{f \in {\mathcal F} (X,Y) : f(X) \quad \text{is totally bounded}\} \\ B_ \alpha (X,Y) & = \{f \in {\mathcal F} (X,Y) : f(A) \quad \text{is } {\mathcal U} \text{-bounded for every } A \in \alpha\} \\ B_ \alpha^* (X,Y) & = \{f \in {\mathcal F} (X,Y) : f(A) \quad \text{is totally bounded for each } A \in \alpha\}. \end{aligned} \] A net \(\{f_ \lambda : \lambda \in \Lambda\} \subset {\mathcal F} (X,Y)\) is said to be finally \({\mathcal U}\)-uniformly bounded on the members of \(\alpha\), if for each \(V \in {\mathcal U}\), there exists a \(\lambda_ 0\), a finite subset \(F\) of \(Y\) and an \(m \in \mathbb{N}\), such that \(f_ \lambda (A) \subset V^ m(F)\) for each \(\lambda > \lambda_ 0\) and for each \(A \in \alpha\). If the last inclusion holds for each \(\lambda \in \Lambda\), then the net is said to be uniformly bounded on the members of \(\alpha\). Similarly, a net \(\{f_ \lambda : \lambda \in \Lambda\} \subset {\mathcal F} (X,Y)\) is said to be finally \({\mathcal U}^*\)-uniformly bounded on the members of \(\alpha\), if for each \(V \in {\mathcal U}\), there exists a \(\lambda_ 0\), a finite subset \(F\) of \(Y\) such that \(f_ \lambda (A) \subset V(F)\) for each \(\lambda > \lambda_ 0\) and for each \(A \in \alpha\). If the last inclusion holds for every \(\lambda \in \Lambda\), then the net is said to be \({\mathcal U}^*\)-uniformly bounded on the members of \(\alpha\). The main results included in this paper are the following: 1. The sets \(B_ \alpha(X,Y)\), \(B^*_ \alpha(X,Y)\) are closed in the topological space \(({\mathcal F} (X,Y), {\mathcal T}_{{\mathcal W}_ \alpha})\), where \({\mathcal W}_ \alpha\) is the quasi uniformity on \({\mathcal F} (X,Y)\) generated by sets of the form \(\langle A,V \rangle = \{(f,g) \in {\mathcal F} (X,Y) \times {\mathcal F} (X,Y) : f(A) \subset V(g(A))\}\) for \(A \in \alpha\) and \(V \in {\mathcal U}\). 2. Suppose \({\mathcal W}_ \alpha^{-1}\) denotes the conjugate quasi uniformity of \({\mathcal W}_ \alpha\), and suppose that \(\{f_ \lambda, \lambda \in \Lambda\} \subset {\mathcal B}_ \alpha (X,Y)\) (resp. \(\{f_ \lambda, \lambda \in \Lambda\} \subset {\mathcal B}_ \alpha^* (X,Y))\) is a net converging to \(f\) with respect to the topology \({\mathcal T}_{{\mathcal W}_ \alpha \vee {\mathcal W}_ \alpha^{-1}}\). Then \(\{f_ \lambda, \lambda \in \Lambda\}\) is a finally \({\mathcal U}\)-uniformly (resp. \({\mathcal U}^*\)-uniformly) bounded net on the members of \(\alpha\). The classical theorem which states that if a sequence of real-valued functions defined and bounded on \(X \subset \mathbb{R}\) converging uniformly to a function \(f\), then the function \(f\) is bounded and the sequence is uniformly bounded on \(X\) follows as a corollary to the discussion.