In the theory of compactifications, Magill's theorem that the continuous image of a remainder of a space is again a remainder is one of the most important theorems in the field. It is somewhat unfortunate that the theorem holds only in locally compact spaces. In fact, if all continuous images of a remainder are again remainders, then the space must be locally compact. This paper is a modification of Magill's result to more general spaces. This of course requires restrictions on the nature of the function.
[CFV] Caterino A., Faulkner G.D., Vipera M.C.: Two applications of singular sets to the theory of compactifications
. Rend. Ist. Mat. Univ. Trieste 21 (1989), 248-258. MR 1154977
| Zbl 0772.54018
[M2] Magill K.D.: The lattice of compactifications of a locally compact space
. Proc. London Math. Soc. 18 (1968), 231-244. MR 0229209
| Zbl 0161.42201
[T] Fu-Chien Tzung: Sufficient conditions for the set of Hausdorff compactifications to be a lattice. Ph.D. Thesis, North Carolina State University.