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Clearly, there are examples for the second question. Each Hausdorff abelian paratopological group (that is, a group endowed with a topology making the multiplication continuous) which is not a topological group (for instance, the Sorgenfrey arrow, that is the real line endowed with the Sorgenfrey topology generated by the base consisting of half-intervals ...


Here's an answer to the mathematical part of the question (namely Q2). An example is the group of adeles. Start from $\mathbf{A'}$ defined as the inverse image of $\bigoplus_p\mathbf{Q}_p/\mathbf{Z}_p$ in $\prod\mathbf{Q}_p$, with $\prod\mathbf{Z}_p$ prescribed to be a compact open subgroup (the sum being over all primes); then ...

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