Given a metric space \((X,d)\), the expression \(H(X)\) denotes the set of all homeomorphisms of \(X\) onto itself and \(H_d(X)\) stands for the set \(H(X)\) with the uniform topology generated by the metric \(d^*\) defined by the equality \(d^*(f,g)=\min\{1, \sup\{d(f(x),f(y)): x\in X\}\}\) for any \(f,g\in H(X)\). If \(W\) is an open subset of \(X\times X\) then \(G_W=\{g\in H(X): G(g)\subseteq W\}\); here \(G(g)=\{(x,g(x)): x\in X\}\) is the graph of the function \(g\). The topology on \(H(X)\) generated by all sets \(G_W\) is called the fine topology and the respective space is denoted by \(H_f(X)\).
The authors prove that for a locally homogeneous dense-in-itself metric space \(X\), the space \(H_f(X)\) is first countable if and only if \(H_f(X)\) is metrizable which in turn is equivalent to \(X\) being compact. They also consider three compatible metrics \(\rho, \sigma\) and \(\tau\) on the space \(\mathbb R^n\) for some natural number \(n\) and study the relationship between the topologies of the spaces \(H_\rho(\mathbb R^n)\), \(H_\sigma(\mathbb R^n)\) and \(H_\tau(\mathbb R^n)\).