## Real Analysis Exchange

- Real Anal. Exchange
- Volume 30, Number 1 (2004), 373-384.

### 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 $B_{\alpha}^{\circ}(X,Y)$.

#### Article information

**Source**

Real Anal. Exchange, Volume 30, Number 1 (2004), 373-384.

**Dates**

First available in Project Euclid: 27 July 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1122482144

**Mathematical Reviews number (MathSciNet)**

MR2127543

**Zentralblatt MATH identifier**

1074.26003

**Subjects**

Primary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

Secondary: 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54C20: Extension of maps

**Keywords**

Baire functions Borel functions Zero dimensional spaces Tietze extension

#### Citation

Shatery, H. R.; Zafarani, J. The Equality Between Borel and Baire Classes. Real Anal. Exchange 30 (2004), no. 1, 373--384. https://projecteuclid.org/euclid.rae/1122482144