Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has no countable dense subset [relative to the subspace topology].

Question

Let $\mathbb R_l$ be $\mathbb R$ given the lower limit topology, i.e. the topology generated by $\{[a, b) \subseteq \mathbb R|a<b\}$.

Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has no countable dense subset [relative to the subspace topology].

Can anyone help me on this? Our professor did not discuss about countable dense subset.

Answer 1

Space $\Bbb R_l$ is called Sorgenfrey arrow and it has some exotic properties, making it a useful counterexample, sometimes it is even called a “universal counterexample”. For instance, although the space $\Bbb R_l$ is hereditarily separable, hereditarily Lindelöf, and perfectly normal, its square $\Bbb R_l\times\Bbb R_l$ is non-normal and contains a discrete subspace $\{(x,-x):x\in\Bbb R\}$ of cardinality $\frak c$.