Geodesic
On $G$-Compactifications
Matematičeskie zametki, Tome 78 (2005) no. 5, pp. 695-709.

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The semilattice of $G$-compactifications of a $G$-Tikhonov space $X$ is studied. The question of what sets containing $X$ and contained in the maximal $G$-compactification $\beta_GX$ must be contained also in all other $G$-compactifications of $X$ is considered. Conditions for $\beta_GX$ to be the completion of the $G$-space $X$ with respect to a natural uniformity (proximity) on $X$ are obtained. Sufficient conditions for the existence of a smallest (minimal, unique) $G$-compactification of $X$ are given. 
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K. L. Kozlov; V. A. Chatyrko. On $G$-Compactifications. Matematičeskie zametki, Tome 78 (2005) no. 5, pp. 695-709. http://geodesic.mathdoc.fr/item/MZM_2005_78_5_a5/

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