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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]].\nA *Bohr compactification* of $G$ is a pair $(bG, b)$ of a compact Hausdorff group $bG$ and a [[continuous homomorphism|9J5PQMG3HJ]] $b \\colon G \\to bG$ with this property.\nFor every compact Hausdorff group $K$ and every continuous homomorphism $f \\colon G \\to K$, there is exactly one continuous homomorphism $\\tilde{f} \\colon bG \\to K$ with $f = \\tilde{f} \\circ b$.\n\n## Remarks\n\nThe Bohr compactification exists for every topological group, and it is unique up to a unique isomorphism compatible with the maps from $G$.\nExample: if $G$ is a [[compact|EVHCE5VR6Y]] Hausdorff group, then $(G, \\id_G)$ is a Bohr compactification, since each $f$ factors as $f = f \\circ \\id_G$ and the factor must be $f$.\nNon-example: $(\\mathbb{T}, x \\mapsto e^{2\\pi i x})$ is not a Bohr compactification of $\\R$.\nThe continuous homomorphism $x \\mapsto e^{2\\pi i \\sqrt{2} x}$ from $\\R$ to $\\mathbb{T}$ does not factor through $x \\mapsto e^{2\\pi i x}$.\nA factor $\\tilde f$ would give $e^{2\\pi i \\sqrt{2}} = \\tilde f(e^{2\\pi i}) = \\tilde f(1) = 1$, which is false.\nIf $G$ is a locally compact abelian group, $bG$ is the [[Pontryagin dual|FMW57WNPDB]] of the group $\\widehat{G}$ with the discrete topology, and $b(x)$ is evaluation at $x$.\nThe map $b$ need not be injective, and its image is dense in $bG$.\n",
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