Enlarging the convergence on the real line via metrizable group topologies (Q2702764)

The author answers in the negative the question of R.~Frič whether there is a topology~\(\tau \) on the real line~\(\mathbb R\) such that NEWLINENEWLINENEWLINE(1) \((\mathbb R,\tau)\) is a metrizable topological group with respect to the usual addition. NEWLINENEWLINENEWLINE(2) If a sequence~\((x_n)\) converges to~\(x\) with respect to the usual topology, then~\((x_n)\) converges to~\(x\) with respect to~\(\tau \), too. NEWLINENEWLINENEWLINE(3) The sequence~\((2^n)\) converges to~\(0\) with respect to~\(\tau \). NEWLINENEWLINENEWLINE(4) If a sequence~\((x_n)\) converges to~\(x\) with respect to~\(\tau \) and \(a\in \mathbb R\), then the sequence~\((ax_n)\) converges to~\(ax\) with respect to \(\tau \), too. NEWLINENEWLINENEWLINEMoreover, the author shows that there is a topology~\(\tau \) on the real line~\(\mathbb R\) satisfying the conditions (1), (2) and (3).