On \(e\)-compactifications and \(e\)-compactifiable spaces (Q1594585)

Recall some definitions. An extension \(Y\) of a Hausdorff space \(X\) is called an \(e\)-compactification of \(X\) if every open covering of \(Y\) has a finite subfamily that covers \(X\). A space \(X\) is called an \(s\)-completely regular space in \(Y\) if the space \(X \cup \{ y\}\) is completely regular for every point \(y \in Y\). So, \(e\)-compactifiable spaces are intermediate between regular and completely regular spaces. A natural partition of the hyperabsolute \(HX\) of a regular space \(X\) into \(e\)-compact sets is called an \(e\)-partition. The main results of the article under review are as follows: Theorem 1. All semiregular \(e\)-compactifications of an \(e\)-compactifiable Hausdorff space have the form \(HX / R\), where \(R\) is an arbitrary \(e\)-partition of the hyperabsolute \(HX\). Theorem 2. A regular space is \(e\)-compactifiable if and only if there is a discrete complete family of free regular \(H\)-systems in this space. Theorem 3. A space \(X\) is \(s\)-completely regular in each of its \(e\)-compactifications if and only if the largest \(e\)-compactification \(eX\) of \(X\) is a compact space. The author constructs an example of a completely regular space which is not \(s\)-completely regular in some of its \(e\)-compactifications.