The Machado-Bishop theorem in the uniform topology (Q6632945)

The Machado-Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, giving an analogue of Machado's distance formula for bounded weighted vector-valued functions, the author studies the Machado-Bishop theorem for bounded weighted spaces in the uniform topology. Unlike in the weighted space \(CV_0 (\Omega , X)\), one cannot expect that Machado's distance formula holds in \(CV_b (\Omega, X)\) in the uniform topology, due to Hewitts counter-example [\textit{E. Hewitt}, Duke Math. J. 14, 419--427 (1947; Zbl 0029.30302)]. Instead, the author finds a formula like \(d_ {\Omega} (f, A) = \inf_{E\in \mathcal{A}} d_E (f, A)\) where \(\mathcal{A}\) is some ``simple'' z-filter. Some types of the Bishop-Stone-Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are also discussed.