Countable compactifications and a generalized Hahn-Mazurkiewicz type theorem

In this paper, we provide a characterization as to when a locally connected generalized continuum (a locally compact connected separable metric space) has a countable compactification of a certain order type. We then prove that for any two locally connected generalized continua with maximal countable compactifications X and Y of the same order type there is a perfect surjection f : X → Y. This provides one kind of generalized Hahn-Mazurkiewicz type theorem.