On the uniqueness of extensions of ω

ω denotes the countably infinite discrete space. Our interest is whether a homeomorphism f between remainders of extensions W,W′ of ω can be extended to an onto homeomorphism F between W and W′. Here first we note this is impossible for Tychonoff spaces W and W′. On the other hand we have noted in 1995 that it is possible if both W and W′ are compact and metrizable. Using this result, we obtain the second result that the situation is the same for non-compact metrizable spaces if f(KW)=KW′ holds, where KW [KW′, respectively] is the subspace of the remainder consisting of points at which W [W′] is not locally compact. authorʹs opinion this is a property of the space ω rather than of separable metric spaces (see remark towards the end of Introduction).