Geodesic
The Interval [0,1] Admits no Functorial Embedding into a Finite-Dimensional or Metrizable Topological Group
Serdica Mathematical Journal, Tome 26 (2000) no. 1, pp. 1-4

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An embedding X ⊂ G of a topological space X into a topological group G is called functorial if every homeomorphism of X extends to a continuous group homomorphism of G. It is shown that the interval [0, 1] admits no functorial embedding into a finite-dimensional or metrizable topological group.
Keywords: Topological Group, Functorial Embedding 
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Banakh, Taras; Zarichnyi, Michael. The Interval [0,1] Admits no Functorial Embedding into a Finite-Dimensional or Metrizable Topological Group. Serdica Mathematical Journal, Tome 26 (2000) no. 1, pp. 1-4. http://geodesic.mathdoc.fr/item/SMJ2_2000_26_1_a0/