On completely regular spaces for which \(eX=\beta X\) (Q2731794)

Let \(eX\) be the largest semiregular compactification of \(X\) satisfying the following property: for any open cover of \(eX\) there exists a finite subfamily which is a cover of \(X\). It is known, that \(eX\neq \beta X\) for some completely regular space \(X\), therefore the property \(eX=\beta X\) defines a new class of completely regular spaces. In this paper, the author constructs three examples to show that this class is not closed under passage to infinite sums, subspaces and Cartesian products. The author introduces here a notion of countably regular spaces and considers some properties of countably regular spaces.