О вполне регулярных пространствах, для которых $eX=\beta X$

K. V. Matyushichev

Problemy analiza, no. 6 (1999), pp. 46-56

Résumé

Let $eX$ denote the largest semiregular $e$-compactification of an $e$-compactifiable space $X$. In [1] K. P. Hart and J. Vermeer presented an example of a completely regular space $X$ for which $eX\ne \beta X$, thus distinguishing a new class of completely regular spaces having the property $eX= \beta X$. This paper shows that this property is not preserved by sums, subspaces and Cartesian products. A few remarks are made about $eX$ itself. Finally, we introduce countably regular spaces that are presumably intermediate between completely regular and regular spaces. A space $X$ is called countably regular (CR) if it has a countably regular (CR) base, i. e., a base $\beta$ such that for every $U\in; \beta$ there exists a sequence $\{U_{n}\}_{n=1}^{\infty}$ in $\beta$ such that $U=\cup_{n=1}^{\infty} U_{n}$ and $[U_{n}]\subset U$ for each $n\in \mathbb{N}$. Most widely known regular non-completely regular spaces are not CR. Every time there is machinery killing complete regularity it also kills CR. Two questions arise. Does there exist a CR space that is not completely regular? Does countable regularity imply $e$-compactifiability as is the case with complete regularity?