The question is one in Munkres where we are asked to prove the metrizability of RxR in the dictionary order topology.
My attempts of defining a metric seem to falter at the end. As for example, I have tried the standard bounded metric, usual metric etc…etc., but all of them give us open balls which can't be contained in a basis element of the dictionary order topology of the form (axb, axc), where b<c.
How do i proceed??
Best Answer
HINT: Each vertical $\{x\}\times\Bbb R$ is a clopen subset of $\Bbb R\times\Bbb R$ in this topology, so the space is homeomorphic to $\Bbb R_d\times\Bbb R$, where $\Bbb R_d$ is the real line with the discrete topology, and the second factor has the usual topology. $\Bbb R_d$ and $\Bbb R$ are both metric spaces. The product of two metric spaces is metrizable; do you know how to construct a metric for it from metrics on the factors? If not, or if you get stuck, take a look at the second paragraph of this answer.