A characterization of open mapping in terms of convergent sequences (Q2468975)

The paper contains a proof of the fact that a mapping \(f:X \rightarrow Y\) between metric spaces is open (not necessarily continuous) if and only if \(f^\gets (y) \subseteq \limsup_n f^ \gets(y_n)\) for every sequence \((y_n)\) in \(Y\) converging to \(y\). Here \(\limsup_nf^{\gets}(y_n)\) is the set of all points \(x\in X\) such that there exist a subsequence \((y_{n_k})\) of \((y_n)\) and points \(x_k\in f^\gets (y_{n_k})\), \(k \in \mathbb N\), with \(\lim_kx_k = x\).