Path components in the uniform spaces of continuous functions into a normed linear space (Q2862918)

Let \(C(X,Y)\) be the set of all continuous functions from a Tychonoff space \(X\) to a metric space \((Y,d)\) and let \(C_{d}(X,Y)\) denote the set \(C(X,Y)\) with the uniform topology generated by \(d\). In the present paper, the authors prove that if \(d\) is a compatible bounded metric and \(Y\) contains a nontrivial path then \(C_{d}(X,Y)\) is separable if and only if \(Y\) is separable and \(X\) is compact and metrizable. Furthermore, for a normed linear space \(Y\), the authors study the properties of \(C_{\rho}(X,Y)\) and \(C_{\tau}(X,Y)\), where \(\rho\) and \(\tau\) are two different compatible bounded metrics on \(Y\). They prove that \(C_{\tau}(X,Y)\) is path connected but \(C_{\rho}(X,Y)\) is not connected whenever \(X\) is not pseudocompact, and hence these spaces are not homeomorphic when \(X\) is not pseudocompact. They prove also that the uniform spaces \(C_{\rho}^{*}(X,Y)\) and \(C_{\tau}^{*}(X,Y)\), where \(C^{*}(X,Y)\) is the subset of \(C(X,Y)\) consisting of bounded functions, are homeomorphic.