Compact generation does not imply local compactness

The compact set $K=\{0\}\cup \{1/n!:n\ge 1\}$ generates the additive group $\mathbb{Q}$, since every positive integer denominator divides some factorial. With its usual topology, $\mathbb{Q}$ is metrizable and is not locally compact. Nor can it have a nonzero Haar measure finite on compact sets: invariance gives equal masses to all singletons; a positive singleton mass makes $K$ have infinite mass, whereas zero singleton masses make the countable group have zero total mass.

// Target: analysis.bigb