A Remark on Two Extensions of the Disc Algebra and Mergelyan's Theorem

Abstract

We investigate the set of uniform limits of polynomials on any closed Jordan domain with respect to the chordal metric $\chi$ on $\mathbb{C}\cup \{\infty \}$. We conclude that Mergelyan's Theorem may be extended to the case of uniform approximation with respect to $\chi$ on closed Jordan domains. Similar results are obtained if we replace the one point compactification $\mathbb{C}\cup \{\infty \}$ of $\mathbb{C}$ by another compactification of $\mathbb{C}$ homeomorphic to the closed unit disc.