On Products of Property $b_1$
Received:July 13, 2010  Revised:November 20, 2010
Key Words: $\sigma$-product   Tychonoff products   property $b_1$   hereditarily property $b_1$.  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.10671134; 11026081).
Author NameAffiliation
Jianjun WANG School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan 611731, P. R. China 
Peiyong ZHU School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan 611731, P. R. China 
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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 $b_1$.
Citation:
DOI:10.3770/j.issn:2095-2651.2012.02.012
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