Normal spaces and fixed points of Čech-Stone extensions

We present two examples of nice normal spaces X having the property that for some fixed-point free homeomorphism on X its Čech-Stone extension has a fixed point. One of the spaces presented here is locally countable, locally compact, separable, normal, countably paracompact and weakly zero-dimensional. The other one is hereditarily normal and strongly zero-dimensional. Our construction of this example however requires the Continuum Hypothesis. Since for paracompact spaces with finite covering dimension every fixed-point free homeomorphism has a fixed-point free Čech-Stone extension, these results are “best possible”.