The Equality Between Borel and Baire Classes

Abstract

In this paper, we study some properties of the Banach space $\beta_{\alpha}(X)$, which consists of all real Baire functions on a perfectly normal space $X$. We obtain the equality between Baire and Borel classes as a consequence of existence of an approximation property and a Tietze extension for these classes. Moreover, when $Y$ is a zero dimensional topological space, we obtain a refinement of the known results for the equality between ${\beta}_{\alpha}^{\circ}(X,Y)$ and $B_{\alpha}^{\circ}(X,Y)$.