An example of a Lindelöf topological space which is not $\sigma$-compact

Question

I am looking for an example of a Lindelöf topological space which is not $\sigma$-compact.

I have looked in Counterexamples in Topology, but, if I am not wrong, all the examples there which meet my requirements are relatively "nice". I am looking for a space which is not Borel (or even analytic). Does anyone know of an example like that, or, is anyone familiar with another, maybe more contemporary, source for interesting topological spaces?

Thank you!

Answer 1

For me a simple and well-known example of a Lindelöf topological space which is not $\sigma$-compact is the Sorgenfrey line $S$, that is the real line endowed with the Sorgenfrey topology generated by the base consisting of half-intervals $[a,b)$, $a<b$. It is well-known that $S$ is Lindelöf (see, for instance, [Eng, 3.8.11]). But the space $S$ is not $\sigma$-compact, because it’s suare is not- Lindelöf (it contains a closed discrete set $\{(x,-x):x\in S\}$ of cardinality continuum).

[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

Answer 2

A Bernstein set is a set $X\subset\mathbb R$ such that every uncountable closed set meets both $X$ and $\mathbb R\setminus X$. It is Lindelöf because it's a separable metric space; it is not $\sigma$-compact because it has the cardinality of the continuum and every compact subset is countable; it is not very nice because it is not Lebesgue measurable.