Spaces not distinguishing pointwise and quasinormal convergence of real functions (Q1183630)

A sequence of real-valued functions \(\langle f_ n\rangle_{n=0}^ \infty\) is said to converge quasinormally to a function \(f\) on a space \(X\) if there exists a sequence of positive reals \(\langle\epsilon_ n\rangle_{n=0}^ \infty\) converging to \(0\) such that \((\forall x\in X)(\exists k)(\forall n\geq k)| f_ n(x)-f(x)| <\epsilon_ n\). A topological space \(X\) is said to be a QN-space if every sequence of continuous real-valued functions on \(X\) which converges pointwise to \(0\) converges also quasinormally to \(0\) on \(X\). A space \(X\) is said to be a weak QN-space if every sequence of continuous real-valued functions on \(X\) which converges pointwise to \(0\) contains a quasinormally converging subsequence. The authors study these and some related notions. Among their results are \ 1) a characterization of bounded subsets of \({}^ \omega\omega\) in terms of quasinormal convergence, \ 2) estimates of the size of QN- and weak QN-spaces by means of known cardinal invariants of the continuum and \ 3) several results that show that QN- and weak QN-spaces are small both topologically and measure-theoretically.