Uniformly continuous 1-1 functions on ordered fields not mapping interior to interior

Document Type : Research Article


Department of Mathematics, Tarbiat Modarres University, Tehran


In an earlier work we showed that for ordered fields F not isomorphic to the reals R, there are continuous 1-1 unctions on [0, 1]F which map some interior point to a boundary point of the image (and so are not open). Here we show that over closed bounded intervals in the rationals Q as well as in all non-Archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 functions not mapping interior to interior. In particular, the minimal non-Archimedean ordered field Q(x), as well as ordered Laurent series fields with coefficients in an ordered field accommodate such pathological functions.


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