Ends of non-metrizable manifolds: a generalized bagpipe theorem

We initiate the study of ends of non-metrizable manifolds and introduce the notion of short and long ends. Using the theory developed, we provide a characterization of (non-metrizable) surfaces that can be written as the topological sum of a metrizable manifold plus a countable number of long pipes in terms of their spaces of ends; this is a direct generalization of Nyikoss bagpipe theorem.