In message <Pine.NEB.email@example.com>, William Elliot <firstname.lastname@example.org> writes >What more general definitions of Cech Stone compactifications are >there. Any beyond the construction with filters and the construction >for Tychonov T0 spaces?
The one you referred to in your original post, i.e. in Wikipedia. Defined categorically in terms of the universal property and constructed using continuous functions into [0,1]. That's not always a compactification in the usual sense, which is why they added the comments which puzzled you.