Geodesic
Equational Compactness of G-Sets
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 11-18

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This paper deals with the notion of equational compactness and related concepts in the special case of G-sets for an arbitrary group G. It provides characterizations of pure extensions, pure-essential extensions, and equational compactness in terms of the stability groups of a G-set, proves the general existence of equationally compact hulls, and gives an explicit description of these. Further, it establishes, among other results, that all G-sets are equationally compact iff all subgroups of the group G are finitely generated, that every equationally compact G-set is a retract of a topologically compact one, and that for free groups G with infinite basis there are homogeneous G-sets which are not equationally compact. 
Banaschewski, B. Equational Compactness of G-Sets. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 11-18. doi: 10.4153/CMB-1974-003-2
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     title = {Equational {Compactness} of {G-Sets}},
     journal = {Canadian mathematical bulletin},
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     year = {1974},
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     doi = {10.4153/CMB-1974-003-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-003-2/}
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