Are there other topologies on $\mathbb R$ that make it a topological field?
As is well known, $\mathbb R$ with the standard topology is a topological field. It is also not hard to check that the discrete and the indiscrete topology on $\mathbb R$ result in a topological field, simply from the fact that all functions from a discrete topology are continuous, as are all functions to an indiscrete topology.
However I wonder if there are other topologies that make $\mathbb R$ into a topological vector space, in particular other topologies that can be easily written down.
Here's one idea: Let's define a set as open if its intersection with the rational numbers agrees with the intersection of an open set of the standard topology with the rational numbers. This is easily seen to be a topology, however I'm not sure if it makes all operations continuous.
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