= Solution
\b[False.] Give the two-element group $C_2$ the <indiscrete topology> and take $G=\mathbb R\times C_2$ with the <product topology>. The indiscrete factor is a <topological group>, since every map into an indiscrete space is continuous, and the product group operations are continuous. Points $(0,e)$ and $(0,a)$, $a\neq e$, have identical neighbourhoods, so $G$ is not a <Hausdorff space>. However, its continuous projection onto $\mathbb R$ is surjective. If $G$ were compact, its image $\mathbb R$ would be compact, which it is not.
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