Geodesic
On~a~problem of Zambakhidze--Smirnov
Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 3-11.

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We say that the action extension problem is solvable for a bicompact group $G$ if for any metric $G$-space $\mathbb X$ and for any topological embedding $c$ of the orbit space $X$ into a metric space $Y$ there exist a $G$-space $\mathbb Z$, an invariant topological embedding $b\colon X\to\mathbb Z$, and a homeomorphism $h\colon Y\to Z$ such that the diagram $$ 
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S. M. Ageev. On~a~problem of Zambakhidze--Smirnov. Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a0/

[1] Bredon G., Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980 | Zbl

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[3] Antonyan S. A., “Ob odnoi zadache L. G. Zambakhidze”, UMN, 41:5 (1986), 153–154 | MR

[4] Engelking R., Obschaya topologiya, Mir, M., 1986

[5] Pontryagin L. S., Nepreryvnye gruppy, Nauka, M., 1973