Uniformly finer topologies (Q1373417)

The author gives examples of metric \((\sigma\)-compact, Polish, complete and locally compact) spaces \((X,\tau)\) which admit a topology \(\tau'\), uniformly finer than the topology \(\tau\), such that \((X,\tau')\) and \((X,\tau)\) are homeomorphic. He shows that for \(\sigma\)-compact Čech-complete spaces this cannot happen. Further he shows that for homeomorphic metric spaces \((X,\tau)\) and \((X,\tau')\) where \(\tau'\) is uniformly finer than \(\tau\), there exists a nonregular topology \(\tau''\) on \(X\) such that \(\tau \subset \tau'' \subset \tau'\).