On Products of Property $b_1$

Abstract:

      In this note, we present that: (1)~Let $X$=$\sigma\{X_{\alpha}:\alpha\in A\}$ be $\left| A \right|$-paracompact (resp., hereditarily $\left| A \right|$-paracompact). If every finite subproduct of ${\rm \{ } X\-\alpha: \alpha \in A {\rm \} }$ has property $b_1$ (resp., hereditarily property $b_1$), then so is $X$. (2)~Let $X$ be a P-space and $Y$ a metric space. Then, $X\times Y$ has property $b_1 $ iff $X$ has property $b_1 $. (3)~Let $X$ be a strongly zero-dimensional and compact space. Then, $X\times Y$ has property $b_1 $ iff $Y$ has property $b_1$.