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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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