Inequality and equality for the extent of products with a special factor
Keywords:
extent, product, rectangular, almost discrete, monotonically normal, $\sigma$-space, strong $\sigma$-space, $\mathbb{D C}$-likeAbstract
For a space $X$, let $e(X)=\sup \{|D|: D$ is a closed discrete subset in $X\} \cdot \omega$, which is called the extent of $X$. First, we give some examples of a rectangular product $X \times Y$ with $e(X \times Y)> e(X) \cdot e(Y)=\omega$. Secondly, we give an equivalent condition for a given space $X$ such that $e(X \times Y)>e(X) \cdot e(Y)$ for a certain special factor $Y$. Finally, we discuss when $e(X \times Y)=e(X) \cdot e(Y)$ for a product $X \times Y$ with a special factor $X$.
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