Exercise 23, § 1

Let $X, Y$ be two completely regular spaces. Now prove that the three conditions are equivalent: $\alpha)$ $X \times Y$ is weierstrassian; $\beta)$ the canonical mapping $\beta(X \times Y) \to \beta(X) \times \beta(Y)$ is a homeomorphism; $\gamma)$ for every numerical function $f$ bounded and continuous in $X \times Y$ and every $\varepsilon > 0$, there exists a finite open covering of $X \times Y$ formed of sets of the form $U_i \times V_i$, where $U_i$ is open in $X$ and $V_i$ open in $Y$, such that the oscillation of $f$ in each of the $U_i \times V_i$ is $\leq \varepsilon$. (To see that $\alpha$) implies $\beta)$, use Exercise 22 c); to see that $\gamma)$ implies $\alpha)$, use Exercise 22 d)).