Two examples of perfectly normal spaces

We construct a perfectly normal space X locally homeomorphic to the irrationals such that for some fixed-point free homeomorphism f : X --> X its Cech-Stone extension beta f : beta X --> beta X has a fixed point. This example is related to an example of Good [8] of a normal space of this kind and a recent construction of van Hartskamp and van Mill [9]. We give also an example of a perfectly normal space X with indX = 1 no compactification of which has small transfinite dimension - a modification of a normal space with these properties constructed by Charalambous in [3]. In both our examples we apply a construction of perfectly normal spaces given by E.Pol and R.Pol in [16].