Topological properties of Sorgenfrey line.

Question

Consider $\mathbb R$ with lower limit topology (generated by taking $[a,b)$ intervals as basis).The topological space thus generated is called Sorgenfrey line.What are some interesting properties of this line in terms of countability axioms and compactness,connectedness?

Answer 1

The Sorgenfrey plane is an example of a separable space admitting an uncountable discrete subspace (which is thus not separable). Indeed it is hereditarily separable (all subspaces are separable, e.g. the rationals are dense in the whole space) and it is hereditarily Lindelöf, but does not have a countable base.

A proof that it does not admit a countable basis: suppose $B_j$ with $j \in J$ is a base for this space. Then, for every $x \in \mathbb{R}$ there is a $B_{j_x}$ such that $x \in B_{j_x} \subseteq [x, x+1)$. So, if $x$ is different than $y$, say without loss of generality $x < y$, then $x \notin B_{i_y}$ and $ x \in B_{i_x}$, so we get that $B_{i_x} \neq B_{i_y}$. This proves that, since all was arbitrary, we can find as many basis elements as points in $\mathbb{R}$.

Moreover, it is an example of a Hausdorff, perfect, non-metrizable space but First-Countable.

Note that the Sorgenfrey line is also totally disconnected, and you can find a proof here. One can also prove that every compact subspace of the Sorgenfrey line is countable: it is proved very nicely by Henno here.

Here are some other nice properties about it.