Is product space of lower limit topology separable?

Question

$(\mathbb{R}_l, \tau)$ : lower limit topology.

I wonder whether $\mathbb{R}_l \times \mathbb{R}_l$ is separable. In my opinion, it is true since $\mathbb{Q} \times \mathbb{Q}$ is dense in $\mathbb{R}_l \times \mathbb{R}_l$:

(sketch of proof) let $W$ be a open set containing $(q_1,q_2) \in \mathbb{Q} \times \mathbb{Q}$. Then there are basic (open) sets $U=[a, a+r_a)$ and $V=[b, b+r_b)$ such that $q_1 \in U$, $q_2 \in V$. This means $W$ contains another rational pair, namely $(q_3,q_4)$. $\Box$

Please let me know if there are any errors.

Are you using that the lower limit topology is itself separable? If you are, you can use the fact that separability is conserved under the product of a countable number of Hausdorff spaces — Miguel Recio, Feb 09 '21 at 15:12

score 3 · Accepted Answer · answered Feb 09 '21 at 15:30

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Yes, your argument in essence proves the correct statement that the product of two separable spaces is again separable (this even extends to continuum many (!) separable spaces by the Hewitt-Marczewski-Pondiczery theorem, BTW, but that’s overkill here).

answered Feb 09 '21 at 15:30

Henno Brandsma

242,131

1

For anyone curious about the overkill, a sketch of the proof can be found here, along with references. – Brian M. Scott Feb 09 '21 at 19:24

Is product space of lower limit topology separable?

1 Answers1