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You will find much on Čech-Stone compactifications in sections 110-113 of Steen & Seebach's popular "Counterexamples in Topology" and here is a personal reference related to section 58.5:
Jesus Ferrer (problem), A.A.Jagers (solution), Problem 6590[1989,65], American Mathematical Monthly, 97 No.10 p.934 (1990). My first topology book, bought at "de Slegte" in Utrecht, was Čech's "Topological Spaces" [Revised edition by Zdeněk Frolíc & Miroslav Katětov].
As a tribute I quote Theorem 41 D.5. [the main theorem on Čech-Stone compactifications]:
Every uniformizable space has a Čech-Stone compactification, and each of the following conditions is necessary and sufficient for a compactification C of a uniformizable space P to be a Čech-Stone compactification of P:
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A.A. Jagers
In June 2003 we visited the Czech and Slovak republics with, what a pleasure to read, the rough guide at hand. While walking on the main street of the small town of Kežmarok, from the beautiful wooden Articular Church [with a capacity of nearly 1500 attendants!] to the former Thököly castle, my eye fell on a plaquette commemorating Hronec's 16 years as head of the renowned local grammar-school. The other mathematician, on the stamp, is Štefan Schwarz. He wrote a thesis on reducibility of polynomials over finite fields, which should explain the lattice in the middle [his supervisor Karel Petr appears on another stamp, at the top of this page]. Schwarz most important work was in semigroup theory. Some 35 years ago I enjoyed reading the classical book of A.H.Clifford & G.B.Preston on this subject, prompted by my first room-mate in Twente, Anton Jetten.