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Enlarging the convergence on the real line via metrizable group topologies. (English) Zbl 0957.22001
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
(1) \((\mathbb R,\tau)\) is a metrizable topological group with respect to the usual addition.
(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.
(3) The sequence \((2^n)\) converges to \(0\) with respect to \(\tau \).
(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.
Moreover, the author shows that there is a topology \(\tau \) on the real line \(\mathbb R\) satisfying the conditions (1), (2) and (3).
MSC:
22A05 Structure of general topological groups
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54H11 Topological groups (topological aspects)
54E35 Metric spaces, metrizability
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References:
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[2] DOBOŠ J.: On modifications of the Euclidean metric on reals. Tatra Mt. Math. Publ. 8 (1996), 51 54. · Zbl 0928.26008
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