Question: In which of the three topologies does $X$ have a countable basis?
Below is the way I did....Can someone verify my proof ? Let me know if there is any concern or questions.
2026-03-25 11:25:59.1774437959
In which of the three topologies does $X$ have a countable basis?
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$\Bbb R^2$ is separable metric, so all of its subspaces are second countable; these should be standard facts.
In the railroad metric, we again have a metric space that is $\sigma$-compact (as all $S_n$ are compact) and so Lindelöf and second countable. I don't really get what you're going for with your attempt.
Show that at $v=(0,1)$, the space in the coherent topology is not first countable (this can be shown using a diagonalisation argument), so the total space cannot be second countable either.